BMPs are not know for their transparency options. I think you can do it under special circumstances, the image has to be low resolution 256 color indexed and the top corner pixel (0,0) is supposed to be the color you want to be transparent and such but I don't like this option as GIF would be better for that anyway.
GIF and PNG are the two most common formats that support transparency. PNG saves at the highest quality and has the greatest transparency flexibility. Also results in a generally larger file size.
GIF saves at a lower quality and can only define one color as transparent. Usually a smaller file size than a PNG.
So depends on what you want and need.
With a PNG it's simple, select the PNG format from the file format list and hit save. It should save the transparency. If it doesn't, that's fine you just need to make a small adjustment.
Click Save As
Select PNG from the file format list
In the bottom right-hand corner you'll see an "Options" button. Click that, then click "Run Optimizer."
Under the "Transparency" tab select "Alpha Channel Transparency" and "Existing Image or Layer transparency." This adds more information to the image but also gives the greatest transparency ability.
Or you can select "Single color Transparency" which will use a very specific color within a given tolerance to create the transparency. You can then set a specific color or tell it to use the current image transparency.
When you save out from now on, it'll use the transparency options that you selected.