Capture the Flag   Capture the opposing team's flag and bring it back to the player's team's flag. Points will only be added if an enemy has not got hold of the team's flag.
Team Deathmatch   Two teams of four to six players face off against each other in a match to the death. The first team to reach 75 kills within the 10 minute time limit will win. If the time runs out, the team with the most kills will win.
Kill Confirmed   Objective and deathmatch. Killed enemies drop dog tags. Team with most dog tags wins.
Search and Rescue    Two teams battle and drop tags when killed, the difference being in Search and Rescue obtaining a teammates dog tags will revive them while if an enemy grabs them, they will be eliminated from the round.
Momentum  

Like War from World at War, five flags or capture points are placed on the map. When the game begins, the central capture point is active, and whichever team captures it moves the active capture point towards the enemy side, where the process repeats. The game ends when one team successfully captures the last point on the line.

Uplink   The objective is to bring a satellite drone, which is placed in the middle of the map, to the enemy uplink station. Each team can score 1 point if the drone is thrown in or 2 points if the player exo boosts into the uplink station with the drone. Teams switch sides at halftime, and whoever has the most points wins the match.
Hardpoint   Teams must rush to secure a "hardpoint" on the map and repel the area from the enemies. Holding the hardpoint increases the teams' score, but if no players are in the hardpoint, no points are gained.
Search and Destroy   An attacking side to either eliminate the defending team or detonate either one of two bomb sites. Players only get one life per round, with most versions of the mode going to a best-of-seven rounds (first to four rounds wins). There is an intermission/half when two or three rounds are completed.
Domination   Own all of the control points on the map.To capture a control point, a player must stand near it for 10 seconds with no enemies nearby to capture it for the player's team.
Infected   The game starts with all players being survivors. In the upper left hand corner of the screen, there is a timer that starts with nine seconds counting down to the Infected. After the nine seconds, one player is randomly chosen to be first infected.  The survivors are the players who are not infected, hence the name "Survivors". The surviving players must hide from and kill the Infected to try and survive until time runs out; each game lasts five minutes. The Survivors spawn with an unknown weapon (depending on what the custom settings are) and must survive by killing the Infected or hiding. 50 points are given to Survivors every time a Survivor is Infected.
Free-For-All   Deathmatch, every player for himself.
Ground War   A playlist for 12-18 players. Domination and Team Deathmatch.

 

Other sources of information
http://www.callofduty.com/advancedwarfare http://store.steampowered.com/agecheck/app/209650/
http://en.wikipedia.org/wiki/Call_of_Duty:_Advanced_Warfare http://www.gamespot.com/call-of-duty-advanced-warfare/