Does gravity affect time?
Yes. For example, clocks tick more slowly near a massive object than farther away. In other words, if you had two identical very accurate clocks and placed one in a laboratory at sea level, and the other on a high mountain, you would notice that the one at sea level is running more slowly than the one on the mountain.
You might worry that something had happened to the clocks, so you could switch their positions, and now you would find that again the one at sea level is running more slowly than the one on the mountain. Don't imagine that you can use this to, say, put off that next birthday. After a few billion years, the core of the Earth is a couple of years younger than the surface. But it is not too small an effect to measure. It was first confirmed experimentally in 1959 by Pound and Rebka.