Equivariance is a property of a function or model: when the input is transformed in some way, the output transforms in a corresponding, predictable way. A classic example is the convolution in image networks, which is translation equivariant. If you shift an image, the feature maps shift by the same amount. This is distinct from invariance, where the output does not change at all under the transformation; equivariance preserves the transformation rather than discarding it.
The reason equivariance matters in deep learning is that it is a powerful inductive bias, a piece of prior knowledge built directly into the architecture. The formal idea was crisply demonstrated for richer symmetries by Taco Cohen and Max Welling in their 2016 work on group equivariant convolutional networks, which extended convolution to be equivariant to rotations and reflections, not just shifts. By guaranteeing that the network already understands these symmetries, you do not waste model capacity or training data learning that a rotated cat is still a cat.
Equivariance became the organizing principle of geometric deep learning, the broad effort to design architectures around the symmetries of their data. Convolutional networks, graph networks, and equivariant models for 3D point clouds and molecules can all be understood as enforcing equivariance to different symmetry groups: translations, permutations of nodes, or rigid motions in space.
For a general reader, the practical payoff is efficiency and reliability. A model that respects the natural symmetries of a problem needs less data, generalizes better, and behaves consistently. In high-stakes domains like medical imaging or molecular design, this consistency, knowing that a result will not flip just because an input was rotated, is a real safety and trust benefit.