The error at the top could mean a number of things, but it boils down to the login servers are not contactable.
This could either be a problem at the servers (unlikley as lots of people are logging in) or with your local computer and/or network.
Quick and easy (although I take no responsibility for anything that happens because of this) turn off your firewall software and maybe your antivirus too, then open the launcher... See if it can see the login servers then.
If it can close, restart the firewall/AV and work out what part is blocking it.
If not then there should be more detailed instructions on the support pages to look into what needs to be configured.
I had the same problem a week or so ago, but it sorted itself out after the initial rush of people trying to download it.
How did these tests go for you?