A soft fork is a change to a protocol which is backwards compatible and does not require all nodes to update. A Miner-Activated Soft Fork (MASF) is a soft fork which is activated by miners who begin mining blocks under the updated ruleset. The process for implementing such an update involves miners signalling their support for the new rules. If a critical portion, usually upwards of 90%, of miners signal support, miners are allowed to begin implementing the new rules.
Miners are important members of the network because they produce blocks and process transactions, but they do not determine the rules of the network. If miners implement rules with which nodes do not agree, their blocks will be ruled invalid, exposing them to large financial losses.