What are the factors that we should take care while designing the logo of a site?

If you are referring to how the logo looks then you need to think about your audience and their possible cultural norms and background.
So things like color can invoke certain moods or signify things and events. E.g. the colour yellow in most western countries means happiness or cowardice but in Japn it means courage, mexico mourning and India commerce.
If the logo is going to include slang or unusual words check what they mean in different languages - there have been some spectacular mistakes

This is a good point

make it memorable and very nice... remember... your logo is the ribbon on a present :)

the type of format. do you want it in one single size? jpg might be enough. do you want it in several different sizes, yet just keep the one image? vector comes in handy.
do you want the background of the image to be transparent? ...

and, for the design itself: make sure you get it right the first time.
the logo basically is something you want to use as long as the site is in use, changing it every few weeks because "hey this looks nicer" might make some users think there are some major changes on the site, for instance, the administrator changed, ...
your logo represents the organization that owns the site, and is part of the sites identity. make it stand out in a crowd, don't overdo it, and, even when major visual changes to the page are implemented, keep an identical logo.

Logo should Be According to What Services you Provides . Colour- slection is Main Factor That makes Logo more Professional . A Logo Design Becomes Identity for Buisness Website .

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