Disclaimer: There is no conclusion to this question as of now and given the trouble we have with defining what exactly is conscioussness in humans, not even speaking of animals, I doubt there will be a definite answer to this question anytime soon. My answer will therefore ignore certain philosophical nuances and will work with assumptions that are based on facts, instead of working with pure facts themselves.
Answer:
- presence of cerebral cortex or of a similar structure in other classes of animals than Mammalia
This will be a rather long (and I mean long) introduction concerning the composition of brains in humans and animals and their comparison. It is not crucial for getting the gist of it and it will have its own TL;DR at the end so feel free to skip to there.
Cerebral cortex is a part of mammalian brain to which things like memory, attention, perception, awareness, consciousness or thoughts are linked. The biggest part of cerebral cortex in humans is the neocortex - its evolutionary most recent part.
There can be many things to be looked at when looking at a brain as whole. We can look at the total number of neurons, at the size of brain relative to the body size, at the number of neurons in certain areas (such as the ratio of neurons in neocortex compared to total number of neurons in the brain), but also at things that may be less obvious at the first sight such as the ratio of neural and non-neural cells in the brain, the encephalisation quotient or the neural density of the brain. I will try to address each one first and then draw a conclusion about human brain afterwards.
Looking at the total number of neurons, elephants have much bigger brain in terms of total number of neurons than humans with ~257 billions of neurons in elephant's brain compared to ~87 billions of neurons in human brain. We do not see elephants solving quadratic equations with a swing of their trunk though and so it is clear that simply the total number of neurons alone, while indicative of computational power, is not a good measure of cognitive abilities.
Should we take a look at the brain and body size relationship instead? Not really. While we as humans do score rather highly for this criteria (brain size is 2% of total body mass), even better so do marmosets with their 2,7%. Marmosets are small primates and they do not belong to the brightest of primates. It does not seem likely brain and body size ratio is the answer.
Could we explain this with encephalization quotient instead? Encephalisation quotient is a ratio between actual brain mass and a predicted brain mass for an animal of given size. It takes into account allometric effects. In this measure, humans do, finally, come first. There are problems with this approach, however. If we were to use this measure, rhesus monkeys should be smarter than gorrilas which, based on their behaviour, they are not.
Or is it that the size does not matter in this case? Probably. For example because it does not account for things such as neural density. Neural density, for example, may be one of the reasons why birds are acting intelligently while having such small brains. This study determines that birds are able to pack twice as many neurons as primates to the same brain size.
If size is out of the picture, what else could we look at? Where human and elephant brains differ notably is for example where these neurons are located. For humans, the ratio of cerebral cortex neurons is much higher, with ~16.3 billions of neurons in their cerebral cortex, compared to only ~5.6 billions of neural cells in the cerebral cortex of elephants. Is this the defining characteristic for human like cognition? And if so, what about great apes whose brains are similarly constructed with rather high numbers of neurons in the cerebral cortex? Although human brain was for long believed to be unique and somewhat outlier amongst animal brains, this notion is gradually being debunked as can be seen for example here or here and the studies show that human brain is simply a scaled-up version of primate brain. For gorillas and chimpanzees for example, the aforementioned ratio is even higher than in humans, since of the total number of ~33 billions and ~22 billions of neural cells respectively, gorillas have ~9.1 billions of these in their cerebral cortex and chimpanzees ~6 billions.
As recent research shows, there are some differences among the brains of other primates and ours. One such measure is the neuropil space.
The neuropil is defined as the space between neuronal and glial cell
bodies that is comprised of dendrites, axons, synapses, glial cell
processes, and microvasculature. Accordingly, the proportion of
cortical gray matter that is composed of neuropil may serve as a proxy
for the total interconnectedness among neurons from local intrinsic
circuits and extrinsic projections within a region.
Just as an intersting fact, the optimal proportion has been calculated to be 3/5 of the gray matter.
To continue
.. it has been reported that there is a significantly greater fraction
of neuropil space in layer III of the frontopolar cortex (area 10) and
Broca’s area (areas 44 and 45) of humans relative to apes, while there
are no phylogenetic differences in the neuropil fraction among primary
sensory and motor cortical areas
Broca's area is an area related to speech production and Area 10, while not very well understood yet, is indicated ot be relevant in strategic thinking, memory recall and other executive functions.
The study concludes that
.. our findings suggest that significant modifications of
neuropil within particular prefrontal cortical areas accompanied the
evolution of the human brain. Such anatomical specializations might
potentially provide increased capacity for enhanced neuronal
integration to support executive cognitive functions.
Since our knowledge about the functions of the cerebral cortex shows that it is the centre of abilities we usually associate with consciousness and higher cognitive processes, we could simply stop here and conclude that what is needed to achieve human-like cognition is a certain number of neurons (with there being a non-linear relationship between the number of neurons and cognitive abilities, hence our much better cognitive abilities comapred to other priamtes) with a certain proportion of these neurons located in cerebral cortex (this being the reason why elephants are not an intellectual master race) and with a certain proportion of cortical gray matter being composed of neuropil (giving us advantage over other primates).
However, things are not so simple still. This conclusion would exclude animals which we know pass some tests related to consciousness (I will be addressing this later) such as some types of birds or even invertebrates such as octopuses, but which do not belong to the Mammalia class and do not have cerebral cortex at all. To explain this, we could look for similar structures in non-mammalian brains, perhaps based on the genetical similarities.
This can be hard.
Whether particular brain structures in different species are related
to each other (i.e., whether they diverged from a single structure in
a common ancestor) is often a subject of debate and controversy. It
can be difficult if not impossible to determine based solely on
location, anatomical organization or functional similarity. This is
because it is relatively easy for these parameters to change over the
course of evolution – they can be affected by changes to one or two
genes, which means there is plenty of variation in these phenotypes
within the population – the raw material for evolution by natural
selection.
But luckily:
If the final phenotypic end-point of any particular region is quite
variable, the opposite is true of the genetic pathways that specify
the identity of the region at earlier stages of development. These
involve master regulatory genes, whose expression is turned on or off
in various parts of the embryo...
...These genes interact in a complicated network of feedforward and
feedback loops to orchestrate this complicated sequence. The networks
in which they operate are so interlocked and involved in so many
different parts of the embryo that mutations to any one gene tend to
have very drastic consequences and will be rapidly selected against.
These early regulatory systems are thus far less variable and tend to
be highly conserved across evolution...
It is thus possible to tell whether a brain region in one species is
homologous to one in another species (which may look quite different
in its mature characteristics) by looking at how those regions were
specified.
And after the introduction, the interesting part:
Similar patterns of gene expression argue that the cortex of mammals
and the “pallium” of birds and reptiles are indeed related to each
other.
And even more interestingly, the study mentioned later on in the article has found out that the commonalities go as far as mushroom bodies of a certain annelid worm, called Platynereis.
It is therefore not only possible, but highly plausible, that even species which do not possess cerebral cortex may be capable of the same or similar things as those who do (and as we shall see later, they do seem to be capable of these).
If you have read my answer up until here, you may wonder what all this has to do with the presence of consciousness. I have included much of the previous information for two reasons. It is, I believe, insanely interesting and, more importantly, it shows just how much there is we have to take into account if we want to try and biologically determine the likelihood of a certain animal having consciousness.
Now, does the presence of cerebral cortex really shows that other animals are conscious? As I mentioned in the disclaimer, we cannot know. We can suppose though. And based on what we know about cerebral cortex, based on what we know about brain structure and based on what we know about the genetic origin of similar structures in other classes of animals than mammals, I would venture to claim that we can assume that many animals possess at least some level of consciousness.
TL;DR:
Studies have shown that cerebral cortex, found in mammal's brains, is heavily involved in higher cognitive processes which we usually associate with consciousness. Comparing human brain with that of other mammals, there is no single defining factor that would make ourselves cognizant - and, by its lack, render other animals incognizant - that we know of. This goes, however, beyond the Mammalia class, based on genetical similarities in brain formation etc. it is highly possible that even animals without cerebral cortex (and therefore - if we just had a simple look at its functions and assuemd its lack will signify the lack of these functions - without higher cognitive abilities) do have some similarly performing structures in their brains which may indicate consciousness.
And to add some further credibility to my conclusions, here is the conclusion from the "Consciousness in Humans and Non-human animals" conference:
Convergent evidence indicates that non-human animals have the
neuroanatomical, neurochemical, and neurophysiological substrates of
conscious states along with the capacity to exhibit intentional
behaviors. Consequently, the weight of evidence indicates that humans
are not unique in possessing the neurological substrates that generate
consciousness. Non-human animals, including all mammals and birds, and
many other creatures, including octopuses, also possess these
neurological substrates.
Attendees of which also said that:
The absence of a neocortex does not appear to preclude an organism
from experiencing affective states,
- animal passing a test conerning a function that we usually link to the consciousness
Since this answer is already long enough and there are others who talked about this point before me (great answer was given by Alexei), I will just shortly describe this.
Another way we can determine if an animal is conscious or not is to try and define what we think conscious means and devise tests for animals to pass in order to peruade us of their consciousness. This has been done many times with many animals, with really interesting results.
I may change this later but as of now, I will simply link to some interesting findings and studies for some animals (I really love the octopus ones). I will also add some more pointers later.
Birds:
Planning for the future
Making connections between phenomena
Octopuses:
Article 1
Article 2