What is the purpose of a coffee press? At one of my local cafes, you can order your drink in a coffee press. It looks like a large mug with a pump on top:



What is this used for? Does it affect the taste or strength of the coffee?
It's a French press.

Put ground coffee inside it, pour hot water, stir a little bit, and then wait for 2 to 4 minutes (or longer if you want a strong and bitter coffee).

After waiting, use the pump to filter the ground from the coffee and pour the coffee into your mug.

I recommend using coarse or medium ground coffee, otherwise, the coffee will go through the filter and will end up in your cup (and mouth).

Some extra information here.