It is most commonly seen as match play: A four ball match is a match in which two sides, each consisting of two golfers, play each other using better-ball scoring (each golfer in the match plays his or her own ball throughout - four balls in play, hence the name).
Let's say Side X consists of Players A and B, and Side Y consists of Players C and D. On the first hole, Player A scores 5 and Player B scores 4. Side X's score for Hole 1 is 4 (its low ball). Player C scores 6 and Player D scores 5, so Side Y's score on the hole is 5. Therefore, Side X wins the first hole of the match, 4 to 5.
As noted, the term "Four Ball" can also apply to a stroke play competition. When played as stroke play, four ball sides again consist of two golfers per team. But in the stroke play version, each side is playing against the field. On the first hole, Player A and Player B score 7 and 6, respectively, so the team's score is 6. Continue in that fashion, marking down the team's score on each hole, and adding up the total at the end for the 18-hole team score. Stroke play four ball is probably more commonly called some variation of 2-person best ball in club and association tournaments.
Handicap allowances for various Four Ball competitions can be found in the USGA Handicap Manual, Section 9-4 (www.usga.com).
Alternate Spellings: The USGA and R&A use "four ball" - two words - as the spelling. However, it is more common to see it spelled as one word - fourball. A hyphenated spelling - four-ball - is also common. All are acceptable.
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