I am using illustrator to create the logo and my client want to use it in a video and I want to know the best format to save the file in.
I would only recomend a file format that supports transparency.
This would reduce the list to PNG and PSD.
Export your file as RGB, use the sRGB color profile. Export 2 versions, one with the color profile embeded and one without color profile.
Regarding the size, will depend on how it will be used. If it is for something like superimpose it on a video, if you think it would be used on a Full HD video, make an artboard 1920x1080px and choose the correct proportion and size (for example 400x400px).
But if yhey need it for some animated intro, you probably need to export it in a much bigger size, ErickP recomendation could work. 1920 px width to put an example.
Some video programs like after effects accept native AI files. Flatten every effect and layers and that format could also work.
I'm sure most video editors are similar but I know Final Cut Pro X accepts the following: BMP, GIF, JPEG, PNG, PSD (static and layered), RAW,TGA and TIFF . I would have the client advise in regards to size. You can just save it out with a pixel width of 1920px, which will fill up their entire screen (assuming they are editing in 1080p). This way they can just resize as they wish.