You pour it back into the container it came in when it's cool, and then you throw it out, or take it to your local household recycling centre where they can dispose of it for you.

I pour it into mason jars. If it's a fat type that can be rendered into lard I do that, but for oils and other unsaturated fats they just get stored until the jar is full and then tossed in the trash.

You have a kitchen so tiny it cannot find room for a single old tomato sauce jar? My condolences.

Ziploc bags work for used liquid fats (after they've cooled sufficiently)

I put a plastic bag in a beaker (to hold it), and pour the fat in once it's cooled down. Tie a knot, throw in trash.