Once you have come up with some ideas for domain names on paper your next step is to see if they are available. Unless you have a very unique company name or service you will find that many of your first choice domain names are already owned by someone else. So the next step is to create an acceptable variation that is not already taken.
Warning: Do not go to just any registrar like GoDaddy.com, Register.com or NetworkSolutions.com and start typing in possible domain names unless you are prepared to purchase them that day.
Why: Some of the registrar companies keep track of domain name lookups people done. Some people/businesses get access to this information and may purchase a domain name that you were very interested in but had not purchased.
I have had this happen to me on more than one occasion with very unique domain names. I did some research on various possible names for a customer one day. Emailed the customer the list of available domain names. The next day I went to purchase the chosen name only to find it had been taken sometime during the night.
Domaindiscover.com is a registrar that I have never had this problem with. It is the place I go when trying to find a suitable domain name for myself or a customer.