Making badges is not free. Each badge slot (20x20) is 100,000 credits. Basically you purchase or make your own badge image and then transform it into a badge by buying a slot and submitting it.
Badges can be autogrant (anyone can request them/have them) and displayed in your homepage. If the badges are private then you can only get them if you pay a certain amount of credits for the owner of the badge to grant it to you. Usually you just send the credits to them or buy a sticker and they'll grant it to you.
When you are making your own badge, you can either self fund it (pay for the slot with your own credits) or accept donations (have users send you credits until you have enough for the badge slot and grant them the badge once made).
This is just a simple summary, feel free to ask any questions if you need anything clarified