The technical way of saying it is that if the anion is a stronger base than water, then the solution will be basic. If the cation is a stronger acid than water, then the solution will be acid. If the cation and anion are equally strong, the solution will be neutral. Technically, I think that's tough to follow. Here is the EASY way to do it.
Look at the salt and convert the salt to its corresponding base and acid (or acid and base). Like this.
NaC2H3O2 is sodium acetate and it MUST have come from NaOH (the base) and HC2H3O2 (the acid which is acetic acid). You can see that all I've done is to split HOH with the OH (the negative charge) goes with the cation (the positive charge) to form NaOH while the H (the positive charge) went with the anion (the negative charge) (to form HC2H3O2).
Now, you know NaOH is a strong base. You know HC2H3O2 is a weak acid. Therefore, the solution will be basic (from the strong base). Purists won't like this explanation BUT it works every time. You don't know the strong acids and strong bases? No problem. Look in the back of your text and find the Ka and Kb tables for weak acids and bases. If it isn't listed, the probability is very high that it is strong. Of course, you could have a poor table but most of the common ones are listed in a decent set of tables. I
NH4Cl would be acid BECAUSE NH3 is listed in the Kb tables and HCl is not listed anywhere. So NH3 is is a weak base, HCl is a strong acid, and the solution of NH4Cl is acidic.