From Halopedia, the Halo wiki
Slayer is a popular and common gametype in the Halo video game series, and a staple of all Halo first person shooters. The object of the game is simply to kill as many opposing players as possible. Slayer is an extremely versatile gamemode, and can be played in teams of 2-8 players ranging up to 12-16 players.
Halo: Combat Evolved
Halo 2: Anniversary
Halo 5: Guardians
Infinity and Legendary Slayer
A Halo 4 Slayer variation, with several notable mechanics changes. Players spawn with customised loadouts (including tactical packages and support upgrades), and are able to call in ordnance. Due to the changes imposed by this mode, Halo 4 later introduced Legendary Slayer, a mode more resembling the Slayer gameplay of prior titles - with preset loadouts and no Ordnance.
Fiesta Slayer is a Slayer variant where all players spawn with random primary and secondary weapons, which range everything from basic loadout weapons to power weapons. In Halo 3, Fiesta Slayer was used in the Cinco de Mayo Double EXP weekend hopper, and was an FFA mode. Halo 4 features team-based Fiesta modes in Team Action Sack. Fiesta Slayer was added to Halo 5: Guardians in the Hammer Storm update. Super Fiesta is a gamemode in Halo 5: Guardians that features all the power weapon REQs that were previously exclusive to Warzone and campaign.
SWAT is a small-team mode designed around fast-paced and tactical gunplay. Players spawn with headshot-capable weapons such as the DMR, battle rifle, Covenant carbine, LightRifle and Pistol, with no additional weapons spawning on the map. Players have no energy shields, and are thus susceptible to being killed in only one headshot.
Hammerzeit ("zeit" being German for "time") is a reference to MC Hammer's song U Can't Touch This. See also Hammer Time.