"Tris" simply means "three" and is used in a number of different contexts. However, in biochemistry it's normally used to refer to the common buffering agent tris(hydroxymethyl)aminomethane. You have to be a little careful, though, as there are other uses of "Tris" in biology, which may not refer to tris(hydroxymethyl)aminomethane. (For example, there's two other buffers called "bis-tris methane" and "bis-tris propane", each of which have very different properties from tris(hydroxymethyl)aminomethane.)
Your situation is referring to the standard tris(hydroxymethyl)aminomethane, though. However, when specifying a buffering system, it's often insufficient to specify just the buffering agent used. A buffer consists of a weak acid or weak base in near-equal concentration to its conjugate base or acid. Tris is the weak base being used, but to have a decent buffer you need to also have the conjugate acid form.
As this conjugate acid is charged, there's an obligate counter ion. Different buffering systems may use different counter ions. That's what the acetate is doing here - acting as a counter ion for the conjugate acid of tris(hydroxymethyl)aminomethane.
That's what "Tris-acetate" is specifying. It's using a Tris buffering system with an acetate counter ion. This buffering system is normally obtained by using enough Tris base to get the desired concentration, and then slowly adding enough acetic acid to obtain the desired pH.
Alternatively, you could use something like the Henderson–Hasselbalch equation to calculate the appropriate proportion of base and conjugate acid for your desired pH, and then specifically mix those two forms together. This isn't typically done with Tris-acetate, as the acetate counter ion form of Tris isn't commonly sold. However, Tris hydochloride is, so if you had a system which used a chloride counter ion, you could use this approach. (Though in practice you'll want to aim a little on the basic side, and adjust to the desired pH by titration - the numbers which come out of the Henderson–Hasselbalch equation are theoretically pure, and don't account for non-ideal deviations in behavior and things like weighing errors.)