Cultural borders are determined by culture producing buildings in your city. For instance, the Obelisk adds +1 culture per turn. At certain milestones the city borders will expand to enclose a larger area.
If your city is on a border between you and a neighboring Civ, you'll have to have a higher cultural rating in order to maintain control of the tiles that both cities' borders would normally contain. You can hover over a tile to see what percentage control you have over a particular tile.
The more culture your city is producing, the more likely it is that you will push their borders back. Build culture-producing buildings in border cities to maintain control of your tiles. Culture producing buildings have a small purple "note" icon and usually a number denoting how much culture they produce per turn.
Certain World and National wonders also produce culture, and after researching Drama, you can put some of your Civ's economic output towards culture across all cities. Additionally, Great Artists can produce a Great Work in a city, which adds an immediate culture bonus. This effect is sometimes referred to as a "culture bomb" because it tends to push the borders of the city outward rapidly.
You may also be interested in this article on Cultural Victory, which contains a pretty good list of ways to make your Civ produce more culture.