Hash power is covered in full under Hashrate, which explains how it is measured, what drives it and why it matters for a network's security. This page gives a short definition and explains how the terms are used.
In proof-of-work networks such as Bitcoin, miners compete to find a block by running a cryptographic hash function over and over with different inputs until one produces a result below a target. Hash power, or hash rate, is how many of these attempts a machine, a mining pool or the whole network makes each second. It is measured in hashes per second, with prefixes such as terahashes (trillions) and exahashes (quintillions) for large totals.
The two phrases are used almost interchangeably. "Hash rate" usually refers to the measured speed, while "hash power" stresses the share of the network a miner controls, as in "a pool with 20% of the hash power". The more hash power a network has, the more expensive it is for an attacker to gather a majority and rewrite recent history. When total hash power rises, the protocol raises mining difficulty to keep blocks arriving at a steady pace, and large mining pools are watched because they concentrate it.