According to Canon Law, a cleric generally may not hold public office:
Canon 285 §1. Clerics are to refrain completely from all those things which are unbecoming to their state, according to the prescripts of particular law.
§2. Clerics are to avoid those things which, although not unbecoming, are nevertheless foreign to the clerical state.
§3. Clerics are forbidden to assume public offices which entail a participation in the exercise of civil power.
Canon Law is not absolutely binding - it is possible for a Bishop (including a Pope) to grant a dispensation for a particular cause. So it is not impossible, but it is not ordinary.
A cleric may promote political causes which are consistent with Catholic teaching (such as Priests for Life, who promote the pro-Life cause). In the United States, a priest must be careful to not overtly politicize his pulpit - this could jeopardize the tax-exempt status of his parish.