Does eve online have a monthly fee

Yes, though there is a free 2 week free trial (or 3 week if you get a referral link form someone that is an active subscriber).
The fee afterwards is between $15 and $19 per month, depending on if you buy the game time from CCP directly or through an official third party website.  You can also pay for game time with a Pilot's License Extension (PLEX).  This is an ingame item that can be bought with either ISK (in-game money) or real money through CCP.