This is a giant can of worms. The short answer is "Yes. Tons."

The most popular variant, to my knowledge at least, is Chess960. Invented by Bobby Fischer because he felt chess came down more to memorization than outplaying your opponent. The biggest difference is the first rank (Rooks, Knights, Bishops, Queen, King) are drawn from random and placed on the board. This mixes up the order of the board and creates far more variety for opening moves.

Not exactly a chess variant, as it predates chess, but I find people interested in the history of chess enjoy learning about the Tafl games. You can play Hnefatafl online vs a computer here. Note that the rules used are not 100% certain to be the actual rules, but our best educated guess with what information we have on the game. The main difference is that it isn't really chess, but a different strategy game. Your objective is to get the "King" to safety while being completely surrounded by the enemy. One player starts with a rather large disadvantage and must outplay their opponents every turn of the way to stand a chance at winning.

Then there is the Japanese Shogi which is still played in Japan. Unfortunately the wiki would do a better job explaining the differences than me as my understanding of Shogi stops at "it exists".

I'm sure there are more popular ones that are escaping my mind right now...but it's getting late and I need to sleep.

There are various asian chess-like games such as Xiangqi and Shogi that may have evolved from the same root game as western Chess, but I'm not an expert is those and will leave discussion of them to those more qualified. I will instead only discuss variations of western (international) Chess.

There are a few that fall within in the 20 year askhistorians exclusion zone. To get information about one of the most major of these (Fischer Random, aka Chess 960), ask again in 34 days as it's right at the cusp of that zone.

Capablanca chess is an example of one that is old enough that it's free game for discussion. It was proposed by Jose Raul Capablanca, one of the finest chess players to ever live, while he was World Champion in the 1920s. It's played on a 10x8 board and adds 2 major pieces, the Archbishop and the Chancellor.

In chess, a queen is effectively a hybrid of a rook and a bishop, able to move as either. In Capablanca chess, the two new pieces are similar hybrids. The Archbishop can move as either a bishop or a knight. The Chancellor can move as either a knight or a rook. It is played normally, but with 2 extra pieces per side.

Capablanca chess is not as popular as other variants in part because it requires a special board and pieces.

Bughouse chess is a popular variation from the 60's in which four players play two simultaneous games of chess. Each player has a partner who is plays the opposite color on the other board. When a player captures a chessman from their opponent, they pass it to their partner, who may place it on the board as a move (with some limitations). Bughouse is quite popular today.

Losing chess is a variant from the late 1800's that is still played today. The goal of losing chess is to lose all of your chessmen (the king is considered a normal piece) . If a player is capable of capturing on their turn, they must capture, though they may choose which chessman to capture and what to capture it with if there are multiple possibilities. The game is won when one player loses all pieces and pawns. This variation is still commonly played.

There are many, many more variations of chess as well. Progressive chess, Circe chess and Gothic Chess are some examples.

Additionally, there have been variations in chess rules that were not universally accepted initially. Draws currently give each player 1/2 point for the game, but before that was universally accepted, tournaments required that games be replayed. The 1/2 point draw variation has become the standard now. Some clubs and other organizations also used to count a stalemate (no legal moves for one player, but the king not in check) as a loss for the stalemated player instead of a 1/2 point draw. Chess is full of rules variations like this that were not universally accepted initially but eventually became standard. These are probably not what you're referring to though.