Principal component analysis, often shortened PCA, is a simple method of linear dimensionality reduction.
It also has a non-linear variant called non-linear principal component analysis.
PCA is a simple and elegant idea, and there are many ways to reach PCA. You can train a neural network without any non-linear layers, and it pretty much works as a PCA without the “importance” ranking aspect.
When you perform PCA on N features, you can get up to N principal components out. If you performed PCA properly, these principal components will be sorted from most important to least important. “Important” in this context means explaining the most variance (linearly).