Umm, but wouldn't that violate the resisting acceleration rule?

It's not a rule. It is the definition of mass that my high school physics teacher made me memorize in 1970.

Which raises an important point in the discussion of definitions - mathmatical and otherwise - of physical units of measure (apologies to ER if he already covered this - I haven't had time to read all the entries). The point is that as well as looking to the textbook definition, and as well as analyzing the units of measure for clues (feet per second per second - for example), you can also gain an insight by keeping in mind the instrument that measures that quality.

Think scale (the kind you stand on), here, since we are talking about mass. You can use the instrument to measure the gravitational attraction part of the definition (obviously). And you can use the same instrument to measure the 'resist accelleration' part of the definition. Here's how: suspend a weight in a mostly frictionless environment (let's not quibble about details here, to keep this short). You place your bathroom scale against one side of the weight. You apply a known and constant amount of force to the opposite side of the scale. You measure the resultant speed of the mass. You measure the amount of time over which the force was applied. You record what the scale says over that period.

Using 0 as start speed, and using the resultant speed, and time, you can easily calculate the acceleration (A). Plug that in to the F=MA equation along with the known force (F) (note we did not go into how we made the force nor how we knew its quantity - must we?). Then, using the F=MA eauation, you can very simply calculate the mass. And guess what: it is exactly proportional to what the scale says because it measured the mass's resistance to acceleration. And guess what else: it is exactly proportional to what the scale says when you use it to measure that mass's resistence to earth's gravity.

IE, if two masses are attracted gravitationally, don't they exert a pull on each other and thus accelerate each other to some degree. How does THAT make sense with resisting acceleration?

Purists will tell you it is a falacy to think of gravity as a force, like this, but it really does work if you do, so let's go with it. Yes, the bodies 'pull' each other. Yes, they resist that 'pull'. Else they would instantaneously head for each other at the speed of light.

I mean gravitation essenitally means exerting a force on another object in effect pulling it. But pulling it accelerates it, no (even if only minutely)?

Yes, exactly. What interacting masses do IS accelerate each other.

So, mass both resists acceleration, and causes the gravitation that may accelerate other objects toward itself? But in relativity, there's an equivalence of motion isn't there? IE, the observer sees the observed as accelerating toward them while they remain at rest, but the observed, from their frame of reference sees the observer as accelerating toward the observed with the observed thinking they're at rest, yeah? I mean if there's no outside third part to observe and see they're both moving relative to the third, or one is/isn't moving.

Kind of a mishmash here. I can't make sense of it. Sorry.

A note, though: If you are talking about the various speed of light and frame of reference issues, you are talking special relativity. If you are talking about gravity, then you are talking general relativity. When a layman (like me or you) mixes them together and tries to come up with contradictions, then he will likely give himself a headache, and come up with nothing useful.