There are two ways that Religion spreads: The natural spread that the holy city as well as any other religious city generates and the deliberate actions a player can perform to convert other cities.
The natural spread of Religion is fairly straight-forward. Any city that has a majority Religion (more than half of their citizens are following a single Religion) exerts pressure up to 10 hexes from the city. If another city is within that range, and they do not yet have a majority religion, then the pressure will slowly start to convert citizens. If the city already has a majority religion, then it will take two or more cities exerting pressure to start converting citizens. The more cities exerting pressure on the target city, the faster a city's citizens will be converted. You can view your own outward religious pressure, as well as incoming pressure by mousing over the Religion icon on the city banner (if it already has a Pantheon or Religion).
To speed up the spread of your Religion the following options are available to you: You can purchase Missionaries by spending
Faith in any city that already has the Religion you wish to spread as its majority. To purchase one just enter the city screen and you can open the purchase menu by clicking on the purchase button in the lower left. The Missionary will be grayed out if you don't have enough
Faith to acquire it. Once you have a Missionary you can move him into or next to the city you wish to convert and click on the Spread Religion button in his action menu which will normally expend him after two uses.
Great Prophets can spread Religion in a similar fashion but additionally they abolish any other Religions that might be present in their target city. They can perform this action multiple times before being expended.