There are a lot of different RAW file formats, not compatible with each other. The Wikipedia page has a list of them. Some manufacturers have used more than one format.
There are some things they tend to have in common.
Most of them are based on the TIFF file format. The TIFF file format can contain various types of image data and metadata. Manufacturers tend to use the TIFF file format as a container, but include the raw sensor data inside it, rather than including a regular bitmap image. This raw sensor data can only be read by a compatible RAW image editor/viewer.
The raw image data includes digital readings for each active subpixel in the image sensor. Almost all modern camera sensors use the same RGGB Bayer interpolation matrix, which lays out the sub-pixels in a grid pattern where each 2x2 square of pixels has a Red, Green, Green and Blue sub-pixel. While some may read digital values using 12 bits and some may use higher bit depths, the layout is usually the same.
The raw data is often compressed using a common lossless compression algorithm.
RAW files almost always contain an embedded JPEG image as well. This allows for fast previewing of the image on the camera's LCD screen, with the ability to zoom in on detail too. In many cases, there is also a small embedded JPEG thumbnail as well.
RAW files need to preserve the same metadata streams as the camera would need to create the equivalent JPEG - this means that the EXIF/XMP data will also appear in the file somewhere.
The proliferation of mutually incompatible formats has led for the push to standardise, and the DNG format owned by Adobe is one attempt at creating a (comparatively) open, manufacturer-agnostic format that can be shared. However, the effectiveness of such a format is only as good as the manufacturers' support for it. Some cameras do support the DNG format directly, but they are as yet in the minority. In the meantime, open source code for reading virtually any RAW format is readily available, even though RAW files tend to include some encrypted data.
Lots more information about RAW formats is available on the Wikipedia page so I'd recommend it.