A faction in a political party is sometimes described as "a party within a party." It is a group that focuses on one part of the party's agenda or occupies a particular space on the right or left of the party and lobbies for its particular issues and values. Perhaps the best-known and most influential factions have been those which focus on a single issue. This is because they are often able to seize control of the party's policy on that particular issue, generally due to widespread apathy in the rest of the party.
A good example of this type of faction is the Christian Right within the US Republican Party. The Christian Right is a movement rather than a single organization and includes various groups such as "Moral Majority" and "Focus on the Family." However, it has considerable power in ensuring that Republican candidates embrace a faith-based platform, opposing gay marriage and abortion. Many Republicans may well have no strong views on gay marriage or abortion, or may support freedom of choice in both areas, but there are not enough of these people, and they are not sufficiently vocal for the Republican Party to determine its policies based on their views. Hence the Christian Right faction has great influence in this area.
Excessive factionalism is generally regarded as a serious flaw within a party, particularly when the factions fight against one another, as pro-Sanders and pro-Clinton Democrats did in the 2016 presidential election. Extreme members of factions are often more adversarial in their treatment of other factions within their own party than they are with opposing parties.