If you use this as a log the best format is PNG (Background is transparent) You can ask the source file if you need to edit your logo in future like PSD (Photoshop) or AI (Illustrator)
You should use the vector format. In the long run this is the best choice without any doubt. From the vector file you are able to downgrade to whatever popular webformat you need: jpeg, gif, png,...
Gee why not just ask for the designer to save it in a couple of formats... Seems like PSD is a clear winner then as mentioned if you want a smaller logo type size go with jpg or something else... I would think if someone designs it for you they can save it in a few different formats and explain in small detail. Like use this for your Main Banner across entire width of your site at top for this one... Then to send to other sites send this one... For small add or to display on side of your pages or along footers ...get the idea?
If the logo has elements of photography then it will have to be a bitmap based file format so in that case I would get a layered .psd from your designer. If it is a completely drawn logo that does not use any photographic elements I would ask your designer to develop the logo in Illustrator and provide you with a .ai and .eps. The benefits of vector images (ie: Illustrator) is that they are scalable without degrading the quality of the image, .psd's and bitmap based images are not.
For printing always ask for vector format. Ai EPS are the best formats for printing.. You can resize in any size you want..also u can use it for website.. Thanks.
You can ask your designer whether he has designed your logo in photoshop or vector? you can ask him "ORIGINAL Format", he will send all them.
I am a logo designer, and website template was my job few years ago, for logo vector is the best, for other graphics psd.
If your logo Designer Design Logo in Photoshop then Get .PSD and if he work in COREL DRAW Get .CRD File from his as well as JPG. PNG are the necessary formats for website and once u have original files u can easily edit them in future to change colors etc...
Use PSD when you need to preserve layers, transparency, adjustment layers. Another I wolul like to say is .pdd file which is almost similar to psd files.