There are some good customer relationship management software options out there. A lot of them will have free versions for users with a small amount of contacts, which you would probably fit in assuming you are just talking about your own personal network and not a companies. One example is RelateIQ.
If you don’t want to get this involved, you can set up a pretty simple spreadsheet (Google Docs or Excel) to keep track of your clients.
For example, you could make columns for name, email address, phone, notes on how and when you met, and notes about your contact with them. Then, each row would be a new client.
I hope this helps!