There is no ELO tournament per se. You can play in ELO at anytime by simply joining the ELO room.
The tournament you might be referring to is the usual Daily Tournament in which you can play by being in a valid room when it takes place (every other hour).
I'm actually not sure exactly what hour the ELO tournament rotates. There aren't any screens really that give a GMT, nor is there a profile setting for it so really "18:00" doesn't have meaning to me either. I'm curious too:
At what hour (Give a GMT) exactly does a new weekly ELO begin?
At what hour (Give a GMT) exactly does a ELO ban rotation cycle?
In terms of prizes, I know the ELO tournaments are weekly. In terms of ranking, you can play and try to achieve the best ranking for Daily, Weekly, and Monthly if you're one of THOSE types.
As stated by yoyomoer_xD, the ELO tournament is a weekly event.
To participate, one simply must enter the ELO room and have taken part in at least 5 matches.
Sunday marks the end of the ELO tourney, as well as the start of the next one.
So, once you've meet the requirements, even if you haven't done much, at the end of the week, Kate will send you a pm stating you've taken place in the tournament, and you'll receive 150 clintz and 2 credits I believe.
Head to the ELO game rules via the "The Game" drop down list, and you'll find the prizes given to players.