A container is a file format that packages one or more media streams with the metadata needed to play them in sync. MP4, MOV, MKV, WebM and AVI are containers. The compression happens inside, in the codec; the container just holds the result.
What the choice of container actually affects:
- Overhead. Small and broadly similar. MP4 costs roughly 1-2% on top of the media streams; this site budgets the pessimistic end of that range.
- Compatibility. MP4 is the most widely supported by a wide margin. MKV is the most flexible and the least supported by hardware players. WebM is a web-native subset. MOV is Apple's, and is close enough to MP4 internally that conversion between the two is often free.
- What it can hold. MKV will carry almost any codec plus multiple subtitle and audio tracks. MP4 is fussier.
What it does not affect is file size in any meaningful way. "Convert MOV to MP4 to make it smaller" is a common expectation and mostly wrong: if the video stream is untouched, the saving is the overhead difference, a percent or two. Any larger saving came from re-encoding, which is a codec operation that happened to change the container on the way past.
The useful distinction is between remuxing — repackaging the same streams into a different container, fast and lossless — and transcoding, which re-encodes and always costs quality.