Group mining is better known as pool mining and is covered in full under Mining Pool, which explains how pools work, how they pay members and the concerns they raise. This page gives a short definition and explains why the alternative name exists.
In proof-of-work networks such as Bitcoin, a miner earns a block reward only by finding a valid block, and for a small operator that can take years. By joining a group, many miners combine their computing power so that the group finds blocks regularly, and rewards are shared in proportion to the work each member contributes. Members submit "shares", partial solutions that prove their effort, and the pool operator coordinates the work and distributes payouts, after taking a fee. This turns an unpredictable income into a steadier one, without changing the expected average.
"Group mining" is a plain description used in some glossaries, while the industry almost always says mining pool. The term should not be confused with cloud mining, in which customers rent computing power from a company rather than running their own machines. Large pools also raise concerns about concentration of hashrate, since a few operators can control a big share of block production.