What does it mean to be “stable to deformation”?
Images are susceptible to deformations, i.e. afine or arbitrary deformations, such as "melting" effects.

In computer vision algorithms for non-constrained environments it may be desirable for a predictive model to be "stable", i.e. robust, ideally invariant, to arbitrary transformations that are common in that problem. E.g. a digit classifier would benefit from been stable to common paper deformations such as those caused by wrapping and unwrapping a long letter.

It is said that pooling layers insert certain stability to these deformations, this is analysed by Ruderman on his paper Learned Deformation Stability in Convolutional Neural Networks and in Pooling is neither necessary nor sufficient for
appropriate deformation stability in CNNs.

Some features, such as SIFT, BRISK and HoG try to deal with the most common deformations in image (Scale and Rotation). Most of convolution-dependent methods are already invariant to shifting as that is a feature of convolutional filtering itself.