Hello Uniqua! Welcome to the Forums!
Did Marlon give you a Green Slime Egg? You can put that in the Incubator in the top left part of the screen, as
@FairyRing mentioned.
After a of couple days, a Slime will hatch, and... nothing will happen. That's because you need what
@Cuusardo mentioned, male and female Slimes. Males have the...
thing sticking out of their heads.
To get more Slimes, you'll need more
Slime Eggs. There are two ways of obtaining these, one of them is getting them from killing Slimes (after building a Slime Hutch on your farm). Each Slime will have a 1% chance of dropping a Slime Egg of their respective colour.
The other way is to... kill Slimes. Except this time, you're hunting for the Slime item. This can be compacted into
Slime Eggs via a
Slime Egg Press (25 Coal, 1 Fire Quartz, 1 Battery, recipe @ Combat Level 6). They will come out as a random colour, but Green Slimes Eggs will be the most common.
Once you have a handful of Slimes, they will start breeding, and you can claim the rewards! However, what makes the Slime Hutch especially productive is
filling the four trays of water on the right. This can be done with either Sprinklers or by hand.
Every day they have water,
Slime Balls (not Slimes, not Slime, different) will start to appear in areas that don't have player-placed flooring. This lets you gather Slime and incidentally Petrified Slime, without killing any of your precious Slimes. Which is ideal!
Hope this helps, please update us!