Turf War is basic Splatoon. Your goal is to have the most turf covered until time runs up. A great way to get by is to rush the enemy team and kill all of them repeatedly. This mode is fun and isn't bad for any weapon which allows maximum playability.
-Max
Rainmaker is a really fun mode, and has a fun gimmick. First of all, it introduces a weapon that you can't play anywhere else. This weapon is intentionally really good, which makes the mode interesting. Rainmaker is like Capture the Flag, but the flag is a weapon. The Rainmaker itself is similar to an explosher, but you must charge it. Normally this is fine as a weapon, but in Rainmaker you must push to get to the checkpoint. And a chargeable explosher rushing to get to the enemy base as the entire team is targeting you does not sound easy. This means The Rainmaker must receive tons of support, and people must risk their lives to score.
-Max
Zones is very similar to turf war, but the main focus is in the middle of the map, and the time running out isn't required. your goal is to have middle covered for the longest amount of time, not having the turf covered at the end of the game. This mode is kinda bland, but is still fun.
-Max
Tower control almost has nothing to do with covering turf, and relies on getting on top of a tower at mid, and staying on it for as long as possible. As you stay on the tower, it moves along a path that goes straight into the enemy base. If you stay on it for a really long time, then the game will end due to you making it to the end of the path. This mode is great with brellas and any slayer or support weapon, too.
-Max
Clam Blitz is a weird mode. All around the map, there are these clams scattered everywhere. Collecting 8 clams makes an enormous football to appear that you can throw. Your goal is to throw the football into the goal at their base which opens the gate, and then throw as many clams/footballs into it after that. If you fail to throw in these clams, then it will close back up and will require it to be opened again.
-Max/Nathan
Grizzco is so different from other modes that is essentially its own game. Grizzco has no enemy team, but instead has salmon enemies that you and your team must defeat. There are many enemies, and each has its own way of defeating. There is also mini-bosses that you must defeat to collect their golden eggs. Once you have, you throw them into a basket. You must have a certain amount of golden eggs by the end of the game, otherwise you fail. After a few games of salmon run, you may encounter a King Salmonoid, which are really hard bosses that take your teams entire coordination to defeat. Instead of throwing any eggs you receive from bosses into a basket, you throw them at the King Salmonoid, which does a lot of damage to it. Some Kings have weak spots, so aim at those with the power eggs.
-Max
Tricolor Battle is a splatfest exclusive mode, and is weird. it has 3 teams, 1 defense team, 2 attacker teams. At mid, there is a sprinkler-tower that is called the ultra signal. The attackers are supposed to shoot the tower and get points, the defenders protect the tower and make sure that nobody gets to it.
Fizzbang is the exact same as turf war, just whenever you get kills, you get fizzbangs. fizzangs are similar to suction bombs, but they don't have as much kill power or slowness, but cost no ink to throw.
-Max
Tabbleturf battle is like the "joke" mode of splatoon. This mode is a 1v1 and you use cards on a table instead of weapons on a map. The objective is like turf war: cover the most area in your color. You will have 20 cards in a round that each make a shape out of squares. Cover the table in as many squares as you can to win. You can only place squares that touch other squares meaning you can't just place a chunk of squares in the opponents area.