The basic scoreboard system works like this:
There are four things the scoreboard can be set to measure: Player Deaths (deathCount), Player Mob/PvP Kills (totalKillCount), PvP kills (playerKillCount) & Player health (health). You can name a scoreboard whatever, e.g. Fails,Kills, Lives, etc
They are implemented by typing /scoreboard objectives add name (one of above). The objectives is the part of the command to do with the (one of above).
Then, you will need to make the scoreboard visible. You can have it in three areas, a Sidebar (sidebar), or in SMP, the tab List (list) or on a Players name (belowName) by typing /scoreboard objectives setdisplay (one of above) name

Teams are a bit more complicated/difficult to work with. You can name teams (e.g. Team Herobrine, Team Steve) & add/remove players from teams making it good for PvP, Factions, Adventure Maps or anything you might want to team on.
To create a team, type /scoreboard teams list add name displayname. Display name is not important so just do the same name for name & displayname. By typing /scoreboard teams join name player. Player is where you type the username of the player going in that team. By typing /scoreboard teams leave player, the selected player will leave their team.
Typing /scoreboard teams empty name, it will remove all players from the selected team. And, by typing /scoreboard teams remove name, it will delete the selected team. And finally for basics on teams, by typing /scoreboard teams list name, it will show all players in the team.