The Bitcoin halving is covered in full in the entry Halving, which explains the schedule, lists the halvings to date and discusses their effects on miners and supply. This page gives a short definition and explains the two names.
A Bitcoin halving is a pre-set point in the network's rules at which the block reward, the number of new bitcoin created with each block, is cut in half. It occurs every 210,000 blocks, roughly every four years. The reward started at 50 BTC in 2009 and fell to 3.125 BTC after the April 2024 halving. The schedule, written into the software by Satoshi Nakamoto, is what keeps the total supply below the limit described in 21 million bitcoins, with the last new coins expected to be created around 2140. Each halving reduces the income miners receive from new coins, leaving them more reliant on transaction fees.
"Halving" is the general term, also used for similar events on other networks such as Litecoin, and "halvening" is an informal variant. "Bitcoin halving" simply names the best-known case. All of these terms describe the same programmed cut in new issuance.