The Bite-Sized series of articles is meant to take the most powerful cards in Dominion and give a short explanation of what makes them powerful.
When you play Villa, it’s not the most impressive effect, it doesn’t draw any cards and only gives you $1. On the other hand, Villa has a unique effect in that it allows you to return to your action phase when gained; there are some synergies here that allow you to do some very powerful things.
The most common synergy with Villa’s on-gain ability is that you don’t have to worry so much about running out of actions as long as there are Villas in the pile and you can come up with $4 to get one; this is especially strong with draw cards that want the support of a village, as you can build much more aggressively, delaying your village buys until you actually need the actions, and since you usually immediately play the Villa, you don’t lose a buy and get $1 back right away, softening the blow. Villa loves cost-reducers when being used like this, because they further reduce the opportunity cost of gaining a Villa to keep your turn going.
Some more exotic tricks that Villa can enable would be synergy with draw-to-X cards, since you can play your treasures in the buy phase, buy a Villa, then play your draw-to-X card for lots of value. If you’re overdrawing a lot, you can draw your deck but keep extra draw cards, buy additional payload cards in that same buy phase, then draw and play them in the same turn. While Villas are still in the pile, you can use this to explode in growth very quickly, which means that emptying the Villa pile for denial purposes is often a consideration.