What is the difference between an original and a universal charger?

An original charger comes from the same manufacturer as your battery. A universal charger works on the same voltage and with the same plug, but is made by another brand. Both are suitable; a universal charger is usually cheaper.

Can the wrong charger damage my battery?

Yes. A charger with too high a voltage can damage the battery, and a plug that almost fits makes poor contact. Only use a charger with the same voltage and the same plug as your old one. The voltage is on the label of your battery and of your charger.