$
\newcommand\F{\mathbb F}
$We assume that all vector spaces are over a field $\F$, which is typically $\mathbb R$ or $\mathbb C$ in practical applications.
Linear algebra is nice. We like to transform problems into linear algebra problems, because linear algebra is (relavively) easy and tractable.
A problem in multilinear algebra is a problem involving multilinear functions $V_1\times\dotsb\times V_k\to W$, i.e. these sets are vector spaces and the function is linear in each argument independently. It'd be nice if we could turn a multilinear algebra problem into just a linear algebra problem.
This is exactly what tensors do; in fact, this is the only thing that tensors do! Associated to $V_1,\dotsb,V_k$ is an "essentially unique"${}^1$ vector space which we'll call${}^2$ $T$ together with a multilinear map $\otimes : V_1\times\dotsb\times V_k\to T$ called the ($k$-ary) tensor product. Instead of writing $\otimes(v_1,\dotsc,v_k)$ we usually write $v_1\otimes\dotsb\otimes v_k$. We can call $T$ the tensor product space of $V_1,\dotsc,V_k$, but often just say "the tensor product" of $V_1,\dotsc,V_k$. This is not strictly the same thing as the tensor product of vectors $v_1\otimes\dotsb\otimes v_k$, just overloaded terminology you have to live with.
The way $T$ turns multilinear algebra into linear algebra is the following: for every multilinear $f : V_1\times\dotsb\times V_k\to W$ there is a unique linear $f' : T\to W$ such that $$f'(v_1\otimes\dotsb\otimes v_k) = f(v_1,\dotsc,v_k).$$ You can think of $f'$ as the same thing as $f$, just expressed differently.
We call the elements of $T$ tensors. A tensor of the form $v_1\otimes\dotsb\otimes v_k$ is called simple, and all tensors are sums of simple tensors. The above says, then, that a tensor is just "the thing you input into a multilinear map."
That's it. That is the entirety of what tensors are. All the different "definitions" of tensors are mostly${}^3$ different ways of constructing $T$, but the construction doesn't matter. What matters is that $T$ gives us a way to turn multilinear algebra into linear algebra.
From here it will be useful to use the notation from Footnote (2).
Forming tensor product spaces is associative in a natural way (similar to the use of "natural" in Footnotes (1))
$$
(U\otimes V)\otimes W \cong U\otimes(V\otimes W)
$$
as well as commutative
$$
U\otimes V \cong V\otimes U.
$$
(The tensor product map is not commutative in any sense.) We also have a fundamental duality: every linear map $U\otimes V\to W$ can be uniquely identified with a linear map${}^4$ $V\to U^*\otimes W$ and vice versa. In this way, a linear map is a tensor because $V\to W$ is "the same thing" $\F\to V^*\otimes W$ which is "the same thing" as just $V^*\otimes W$. Because matrices represent linear maps, this is what the sentence "matrices are tensors" means.
Notice that $V^*$ is the "input" space and $W$ is the "output". The same tricks above generalize this idea to all multilinear maps $V_1\times\dotsb\times V_k\to W$: such a map is "the same thing" as an element of $V_1^*\otimes\dotsb\otimes V_k^*\otimes W$.
A very common case is with $V_i=V$ or $V_i=V^*$ for all $i$; the above properties mean the associated tensor product space can be written in the form
$$
(V^*)^{\otimes r}\otimes V^{\otimes s}.
$$
A tensor power $V^{\otimes k}$is just the tensor product space of $V_1=\dotsb=V_k=V$. The above space represents all $r$-multilinear functions that return $s$-tensors
$$
\underbrace{V\times\dotsb\times V}_{r\text{ times}}\to V^{\otimes s}
$$
as tensors themselves.
Footnotes
The specific jargon here is "unique up to unique natural isomorphism": for any two possible tensor product spaces $T,T'$ there is a unique isomorphism (bijective linear map) $\varphi : T\to T'$ which is "natural", meaning that the associated tensor products coincide: $\varphi(v_1\otimes\dotsb\otimes v_k) = v_2\otimes'\dotsb\otimes'v_k$.
This $T$ is usually denoted $V_1\otimes\dotsb\otimes V_k$. This is just notation; you are not supposed to "know" what $\otimes$ means beforehand, and this is not the same thing as the tensor product map, just overloaded notation!
In the context of manifolds and/or differential geometry, "tensor" usually means tensor field, i.e. a sufficiently smooth function $F$ on a manifold with values that are tensor products of copies of the tangent spaces.
$U^*$ is the dual space of the vector space $U$: it is the set of all linear maps $U\to\mathbb F$.
