You seem to have been given the raster version of the logo.
Tell them you need the vector version.
The most common type of vector file type for websites is
Some other common vector file types are
The most common raster file type for websites is
Some other common raster file types are
For more information on the difference between Raster and Vector, see What are the differences between vector graphics and raster graphics?
If they don't have the option of Vector files, there are some options for you, such as cleaning up the image in Photoshop and then tracing (auto or manual - with the Pen tool) in Illustrator, although depending on the complexity of the logo, it can be quite annoying.
Updated, because this may not necessarily be such a good idea (for legal reasons) will leave at that until I get further input.