Thanks to ImageJ's extensibility via plugins, it supports a lot of image formats!
The Bio-Formats plugin provides support for over 130 file formats in microscopy, medical imaging, and related life science formats. Bio-Formats is included with the Fiji distribution of ImageJ.
ImageJ has some built-in support for the DICOM format, and there are also plugins for working with DICOM files. See the DICOM page for full details.
The HDF5 update site enables suprport for reading and writing HDF5 image data. Note that HDF5 is a very broad specification; the HDF5 plugin will not support all manner of HDF files, but only those written according to its constraints.
Out of the box, ImageJ has limited support for some video formats such as AVI and QuickTime, and there are also plugins which extend ImageJ's support for videos. See the Video page for details.