Common Law refers to legal systems similar to the UK (Eg the US) where the “Law” is contained not only in legislation passed by the government but also in all the court cases and rulings by Judges. It originated as “everyone knows torching your neighbors house is illegal, there’s many cases of this that have been done and perpetrators punished, no need to actually write it down just yet”. In most modern Common law systems these laws have since been formally codified and it’s normally just the interpretation of them and their method of implementation that’s buried in case law - judges rulings.

Civil Law: Law that relates to disputes between people, that don’t necessarily need a crime associated with them. In the above “don’t torch your neighbors house” there’s a crime there (arson) but there’s also an obvious financial injury inflicted too. It doesn’t do the neighbor much good to just see the arsonist punishes, because they’re still without a house. Civil Law relates to the victim being able to seek compensation - $$$ - from the other person so they can build a new house. It doesn’t need a crime though - eg say you lend a tool to someone and they lose it, they haven’t committed any obvious crime, but they still owe you a replacement tool. Ditto for accidental damage to stuff you own.