I’m not sure if this is the right place for it, as I am quite happy with the modern definition of a tensor. However I’ve noticed that alot of people tend to use the older definition, so I’m interested in how different people define a tensor (i.e. post the definition of a tensor as you understand it).
Are you referring to the mathematical term, or the physiological term? In physiology a tensor is a muscle that tightens or tenses something, such as the tensor timpani, which tightens the eardrum. I’m not sure about the mathematical usage, so I’ll let someone else handle that.
In physics, generally a “tensor” is a multicomponent quantity where the different components transform among each other subject to certain relations with a change of basis vectors. Modern physicists are starting to finally settle into seeing such “tensors” as elements of “tensor product spaces”, which are the tensor product of some number of representation spaces of some group with the diagonal action of the group.
In general, these days I understand “tensor” in one of the the following two contexts (brace yourself):
A “tensor category” is a catgegory C with a bifunctor ? from C×C to C with a natural transformation a[sub]ABC[/sub] from the composite trifunctor (A?B)?C to the composite A?(B?C), an object 1 and natural transformations l[sub]C[/sub] from 1?C to C and r[sub]C[/sub] from C?1 to C. the canonical example of such a category is that of R-R-bimodules over a given ring R.
Given small linear categories C, D, and E, the “tensor product over D” is a bifunctor ?[sub]D[/sub] from the category C-mod-D×D-mod-E (product category of the categories of bifunctors from C[sup]op[/sup]×D and D[sup]op[/sup]×E to Ab, respectively) to the category C-mod-E. That is, an assignment to every pair of bifunctors F and G in the respective categories, of a new bifunctor F?G in C-mod-E in such a way that transforms appropriately under the action of morphisms in C, D, and E. Specifically, if H is another object of C-mod-E transforming the same way under these actions these is a unique natural transformation from F?G to H. In the special case that C, D, and E are rings, this reduces to the notion of the tensor product of C-D- and D-E-bimodules over D.
I’m not real familiar with tensor theory, but as I understand it, a tensor is a generalization of the notion of a matrix. Mathworld has a decent article on the topic.
I just realized I should disclaim that second context a bit. You won’t find more than one or two other people who would give that answer in terms of tensoring bifunctors rather than just the special case of tensoring over a ring. Partly, this is because it’s something I just polished off on Wednesday 
I work in general relativity, so I know from tensors. In my line of work, given a real vector space V, a tensor of type (p,q) is a multilinear map from V[sup]p[/sup] x (V*)[sup]q[/sup] to the real numbers, where V* is the dual vector space. This is pretty much the prevailing few among people who actually work in GR, although there are those who refuse to get with the times. Oh, and there are even physicists who think about tensors like Mathochist does — Bob Geroch, for one.
I have to admit I’d never heard Mathocist’s defintion before, I was actually thinking more along the lines of MikeS’s defintion i.e. a tenor of type (r,s) is a multilinear map of r vectors and s covectors onto the real numbers.
What suprises me that people are still being taught that tensors are defined by the way they transform; in physics knowing the transformation properties are very useful, but it can be very unenlightening to what a tensor is. There often seems to be confusion among people being taught in this outdated style in the difference between covaraint and contravarint vectors.
Perhaps it’s better thought of as a genarlization of a vector, as a square matrices are not just any old rank 2 tensors, but a mixed tensor of rank 2 (though of course the componets of a (2,0) or a (0,2) tensor they can be written as a matrix) and not all matrices are tensors, though I believe the most general class of arrays are sometimes called quasi-tensors.
That’s a special case of exactly what I said
Given a vector space V over R, define V[sup][/sup] as the space of R-linear functionals on V. Now, a multilinear map from V×…×V×V[sup][/sup]×…×V[sup][/sup] to any abelian group is called balanced if the R action on each factor is the same. The tensor product is the universal such balanced product. That is, it’s a multilinear map satisfying the balancing condition into the vector space V?..?V?V[sup][/sup]?..?V[sup][/sup] such that given any other there exists a unique linear map from V?..?V?V[sup][/sup]?..?V[sup]*[/sup] so the given multilinear map factors through the tensor product.
The difference is that you’re considering functionals on tensor products, which is the dual space. Luckily it can be shown over R that you can just dualize term by term.
As for my mention of a group action, in GR the group is generally GL(V); this is all “really” going on in the category of GL(V)-modules. Anyhow, that’s the difference between V and V[sup][/sup]: the GL(V) action. Without that notion of a group action you cannot tell the V tensorands from the V[sup][/sup] tensorands for a finite-dimensional vector space V.
Geez. Now my brain is full.
What the hell ever happened to, “Pan-fried Semen”, “Is there any reason that semen would be bad for my fish?” and other non-brain-filling questions?
Seriously. You people know way too much and you’re going to hurt yourselves. You just watch.
Just keep in mind that so far, all the bizarre questions about semen have come from non-mathematicians. I’m not necessarily saying that means anything, but I’m not saying it doesn’t either.
I was OK up to “transform”.
Why do I keep looking into these math threads? 
I think there should be a new Board rule. All math threads, if not immediately obvious as being such, and hell, even if they are, should be preceded in the subject title with, “Caution: Math Thread”.
I looked at the question in the forum lobby and my immediately thought was, “Tensor, that’s the thing that I wrap around my knee or elbow or what-have-you every time I fall off of, onto, into, or out of something. I own three.” I had no idea this was a math thread.
Personally, I feel that this rule is fully encompassed by the general “don’t be a jerk” rule, but some additional obviousity (new word) couldn’t possibly hurt.
Oh, so mathematicians are jerks?
ten-sor (ten’suhr, -sôr) : - “Oh poor baby, you couldn’t be any “tensor”, let me massage your neck and shoulders for you. Oh … you like that do you? Here let me work the knots out of your back. Mmmm… nice isn’t it? Oh darn, that bra straps in the way…”
Uh-oh.
Tension, apprehension and dissension have begun.
Well, I would of put soemthing to qualify it, but I’ve never heard of the word ‘tensor’ outside of a mathematical context.
Gosh, heaven forbid someone should accidentally read something about math. :rolleyes:
I couldn’t tell if Standup Karmic was kidding or not . . . but it seems no more reasonable to expect a “Caution: Math Thread” label than, say, a “Caution: Art History Thread” label.
I have nothing against math, or mathematicians either, and I knew it was a math thread from the start. It’s just that I came in here hoping for a nice simple layman’s definition I could understand, maybe with a trivial example or two. But then I just barely got through elementary calculus. I should know better.
OK, really simplified answer (probably to the point of being wrong, but whatever ;)):
A rank 0 tensor is a scalar. (Basically, a number.)
A rank 1 tensor is a vector. (i.e something with a bunch of components, denoted by a single index.)
A rank 2 tensor is a matrix. (two indices)
A rank 3 tensor is whatever comes after a matrix. 
(So it has three indices.)
etc.
This leaves out all this business about transformation properties and contravariant and covariant indices and so forth . . . but you did ask for a layman’s definition, after all. For someone who’s not trying to actually do math, physics, etc., I think getting “a tensor is kinda like a matrix” is pretty good. If you want the technically precise but totally over your head definition, there are many people here who are much more qualified to give it than me (see above.)
A tensor is a rule that associates every vector to another vector.
For example, if you know the unit normal vector to a surface in a body, the stress tensor is a rule that tells you the traction vector on that surface. Or, if you know the angular velocity of a body, the inertia tensor is a rule that tells you the torque “vector”. (Torque is not really a vector, but let’s not go there right now.)
I think this is what you were looking for. The trouble with giving a physical definition of a fundamentally mathematical concept is that one always has to add a disclaimer, like “this definition is neither rigorous nor general.” And I’m not a mathematician.