First off, let me say that any teacher who assumes that everyone knows what tensors are is a lousy teacher.
Now, then, I do General Relativity, so I’m rather familiar with the beasts. It can be somewhat difficult to describe what a tensor is, but it’s much easier to descibe how to represent it and how to use it. Generally, tensors are represented by matrices, and if the tensor in question is a second-rank tensor, as most are (the only one I’ve ever seen that was higher is the fourth-rank Riemann curvature tensor), it’s a two-dimensional matrix with rows and columns, like you’re familiar with. So long as you remember to match covarient tensors with contravarient tensors (more on this in a moment), the tensor multiplication rules look just like the matrix multiplication rules.
Now, as to how to use them: The key to realize here is that usually, when you’re working with tensors, you’re not actually working with the tensors themselves, but just the components of the tensors. The components are just perfectly ordinary scalars, and the final answer of any problem involving tensors will always wind up being a number. Since the components are just scalars, all the normal arithmetic rules apply, and you can deal with them normally.
For example, suppose I have two tensors, T and U. Let’s suppose that they’re both second-rank tensors, so that each element is described by two indices (row number and column number, for matrices), and let’s also suppose that they’re both three-dimensional, so that each index has three possible values. For instance, one element of T might be designated T[sup]12[/sup], or T[sup]33[/sup] or T[sup]23[/sup]. (Of course, there’s six other elements of T that I’m not mentioning there.) In general, we might represent a general element of T as, say, T[sup]ij[/sup], and a general element of U as U[sub]m[/sub][sup]n[/sup]. Notice that I have both of the indices for T upstairs, but the first U index is downstairs: Upstairs indices are called contravarient indices, and downstairs ones are called covarient. It’s important to keep track of which is which when multiplying, and you can get one from the other using something called a metric.
Now, let’s multiply these two tensors. We do this by choosing one index from each to contract (such as the second index of T, and the first index of U), and writing them both in their component form: T[sup]ij[/sup]U[sub]j[/sub][sup]k[/sup]. Note that we used the same letter for the two indices we’re contracting, and that one was upstairs and one was downstairs. This operation will get us a third tensor, which we we call V: T[sup]ij[/sup]U[sub]j[/sub][sup]k[/sup] = V[sup]ik[/sup]. The way we get the elements of V is using something called the Einstein Summation convention: Whenever we see the same index upstairs and downstairs on the same side of an equation, we sum over all possible values of that index. In our case, since we said that these are 3-dimensional tensors, there’s three possible values, so T[sup]ij[/sup]U[sub]j[/sub][sup]k[/sup] = T[sup]i1[/sup]U[sub]1[/sub][sup]k[/sup] + T[sup]i2[/sup]U[sub]2[/sub][sup]k[/sup] + T[sup]i3[/sup]U[sub]3[/sub][sup]k[/sup]. Remember, when we have the indices attached to T and U, we’re talking about the components, not the whole tensor at once, so the math is simpler. Now, if we want (for instance) the 1,3 component of V, we can say that i=1 and k=3, so V[sup]13[/sup] = T[sup]11[/sup]U[sub]1[/sub][sup]3[/sup] + T[sup]12[/sup]U[sub]2[/sub][sup]3[/sup] + T[sup]13[/sup]U[sub]3[/sub][sup]3[/sup]. This is just an equation for one number (the 1,3 element of V) in terms of some other numbers (the components of T and U).