Primary election is the term used in America for the elections which will select the two party's presidential nomination.
When you vote in the primaries, you're actually selecting delegates to attend the party convention who in turn vote for the candidate you voted for. So when Bush won Colorado, what that means is that he got most of Colorado's delegates to the GOP convention to represent him. The number of delegates for each state is proportional to the population of that state.
Those available delegates are split up based on how the vote went within the party in each state. The number of available delegates for Democrats and Republicans in each state can be found in the primary schedule.
The primary elections start in January of election year in what is called the primary season. The primaries are designed to give as much democracy as is deemed possible to local politics and are held on Super Tuesday, the second Tuesday of March in election year.