I had a phone interview with one of the employer. She wanted to know more about how the sponsership for H-1 works. How much it costs, what are the process to be undertaken..Unfortunately, I do not know the whole process. Could somebody please help?

You can tell them that you will hire your own lawyer and take care of all the details.

It just costs them about $200 for the fee. It will cost you about $2000+ for lawyer fees.
Your lawyer will then talk with your employer if he needs anything from them.

The sponsorship for H-1B can be filed on April 1st 2007. You can start working on H-1B only from October 1st 2007. the initial approval will be for 3 eyars, which can be extended gradually.
The cost is 190 for form I-129 +500 for anti fraud fee+ 750 or 1500 depending on the company size reciprocal to the number of H-1B employees. If you want to file a premium processing , an addition of $1000 is required.
Generally lawyers charge 750 upto 2500 depending on the law firm and lawyers.