A checksum is a simple way of detecting errors or mistakes in data.
When it comes to Bitcoin, checksums are calculated using a 32-bit portion of a double SHA256 hash (the first four bytes) and are used to detect incorrect addresses, helping to prevent transactions from going wrong due to simple typos. A checksum only uses 32 bits because that is enough to reliably catch accidental errors, such as a mistyped character in an address, while adding almost nothing to the size of the data.
Checksumming is the process of computing and confirming checksums quickly to validate data without requiring extensive processing or third-party involvement.