I have not run across a strict definition of "term", which may be because it is similar to "element" in that it cannot be defined without a circular definition, but here goes anyways.
A term of a mathematical object (such as a polynomial, sequence, equation, etc) is one simple "piece" distinguishable from the rest of the "pieces" in some way (i.e., in location or description).
For a polynomial, $f(x) = (x-\alpha_1)(x-\alpha_2)\dots(x-\alpha_k)$ you could describe each of the $(x-\alpha_i)$ as separate "terms", where in this case all of the terms are being multiplied. It should be mentioned that this usage of the word is less common, and it is more common to refer to these as "factors". See Term or factor? This does not stop some people from using the word this way however and you may very well find this usage in various pieces of literature.
More commonly, with $f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n$, you could talk about the "$x^2$" term, where $f(x)$ is considered a summation of terms. Similarly, you could have $a_0 + a_1 x + a_2 x^2 + \dots + b_0 + b_1 x + b_2 x^2 + \dots + c_0 + c_1 x + \dots$ and talk about all of the "$x^2$ terms".
In both of these cases, you can treat the terms as being elements of a sequence with some operation occurring between them. Likewise, you can talk about the "terms" of a sequence. Such as the $n$th term in the sequence of Triangle numbers is given by $T(n) = \sum_{i=1}^n i = \frac{n(n+1)}{2}$
What connects all of these examples is that there are many "things" visible in what we are considering and each one of those "things" is colloquially what we call a term.
(other examples: "group all of the even terms together", "find the coefficient of the $x^2y^3z$ term in $(x+y+z+1)^{10}$")
Additional reading: mathword: Term
