In regulation soccer, teams are allowed to play with eleven players on each team, one of whom MUST be designated the GOALKEEPER (not goalie).
Most youth organizations play with fewer players on the field at the lower levels. These are called "small sided games." It is not for us to argue the pros and cons of small sided games. The biggest pro argument is that players will get more touches of the ball, in games.
In US Youth Soccer Association (USYSA) here are the suggested numbers:
U6/7 three players per team with NO goalkeeper.
U8/9 four players per team with NO goalkeeper.
U10/11 six players WITH one player designated as a GOALKEEPER.
U12/13 eight players WITH one player designated as a GOALKEEPER.
U14 and above eleven players WITH one player designated as a GOALKEEPER.
There are various types of youth soccer compitition make-ups. There are basically two diferent ones. One is called "RECREATIONAL," and another "COMPETITIVE (add yet others: extremely competitive, by invitation only, etc.)." There are Olympic Development leagues and many more at the various organiztions. As this site is dedicated to the recreational coaches and parents, we won't go into great detail on these.
Competitive soccer is just what it says; it is designed for the more serious soccer player who plays at a higher level and competes to be on the team and for playing position. Recreational soccer on the other hand is more slanted towards the learning side of soccer with fewer pressures. Make no mistake about it, although recreational soccer is designed for fewer pressures, parents and coaches will become so competitive that often the fun part of recreational soccer gets lost.
In USYSA, the MINIMUM amount of time a player must play is 50% of the game. Unless you have a roster that is twice the number of players in a game, all players should play more than 50%.