Other minds: That there are other minds - such as you who are reading this, if you are not the writer of this text - is one of the most basic philosophical assumptions human beings make.
That it is a philosophical assumption can be seen from two philosophical theses that deny it: Solipsism is the thesis that all there is is oneself, and the brain in the vat thesis is that one is (or may be) in fact no more than some experiment of some higher intelligence(s) that maintain one's brain (or a copy of it, for example in a supercomputer) and feed it information to find out how it behaves. (Then there may be these higher intelligences, but all one thinks, including what one believes about other minds, may be a - manufactured, manipulated - illusion.)
Obviously, while either thesis is logically possible in that neither claim is a contradiction, to believe this seriously means that one runs a considerable risk of being declared insane. And although this risk should not frighten the serious philosopher, either thesis is pretty mad because there is not a shred of evidence for it, just as there is not a shred of evidence for an unlimited amount of similar mere logical possibilities, such as that one is nothing but a dream of creatures living somewhere in Alpha Centauri.
Much more interesting are the following two points involved in there being other minds.
A. What is it like to be another?: Although presumably all sane human beings believe there are and have been other human beings, with experiences, feelings, beliefs, desires, fears and hopes much like them, it also is a fact that all experience any human being has is private - no one can feel another's feelings, make another's choices, think another's thoughts, go through another's joys or pains except imaginatively. (Apart from ESP, for which there is hardly good evidence.)
Note this also includes the seeing of colors, the tastes of food, the pleasures of sex a.s.o., for which there is quite a lot of evidence that people react similarly, but not much reason to believe these reactions are identical for two different persons, and considerable evidence this is not so even in simple cases like the seeing of colors.
For some are colorblind, wholly or partially, and even those who are not do not make all the same distinctions in the same ways when seeing the same colored surface. Thus, what one person may see and classify as a blueish green another may see and classify as a greenish blue.
There is a lot of evidence that different human beings - regardless of race, sex, age, apart from illness and insanity - react similarly to many kinds of things in many kinds of circumstances, and this evidence is both direct and indirect. The indirect evidence is medical and biological: Internal organs are much the same, and indeed exchangeable (hearts, kidneys, livers); reactions to poisons and viruses are generally the same; human DNA is very similar from human to human a.s.o.
There are also many similarities with other animals, especially mammals and apes, but also more differences, and indeed it seems highly likely that the minds animals have, in so far as they have them, are species specific: If one has differently built or placed eyes (birds, insects), presumably one sees in a somewhat different way and may be tuned to different wavelengths. Hence even if all species live in one and the same world, their experiences of it differ systematically, dependent on their kind and the sense-organs and brains of their kind
B. Having a theory of mind: Human beings are educated to a large extent on the basis of the assumption that other human beings are much like them, and have feelings and sensations and ideas much like them, which make it possible to understand and interact with others based on the assumption that others feel and think similarly to oneself.
The idea that another human being is in many ways much like oneself, including in feelings and ideas, even though these are private and not directly accessible to others, is for humans older than toddlers much mediated by language and more specifically by propositional attitudes.
Recently, evidence has been found that some other animals, including apes, in spite of the lack of language, do attribute some kind of mentality - experiences, feelings, intentions - to other animals of the same kind. Indeed, this may be inferred from the fact of playing together, which requires that one ascribes a certain kind of motive to each other.
Thus other animals, especially those that are social and play together, presumably have some sort of theory of mind, in that they seem to attribute to each other at least some emotions, intentions and experiences in certain conditions. What this is like is difficult to say with any confidence for non-humans, for reasons explained in the previous section, and due to the fact that so much of what humans think about themselves and others is tied up with language.