Let's say I have 2 IPs for a given domain (round-robin DNS).
If one the IPs becomes unresponsive, will clients try to connect to the other IP? or they will just fail to establish comunication with the domain?
DNS round robin is not a good substitute for a load balancer. The DNS server will continue to hand out the IP of the node that is down, so some of your users will get to your service and some of them will not.
When the client makes the DNS query, the DNS server returns all of the IP addresses associated with that name. The magic is done by the DNS server rotating the order of that list for every query. However, it is up to the application to implement the capability of "walking" through the list until it finds an IP that works. And most applications don't do that.
Windows Telnet, oddly enough, is one such application that is smart enough to walk the linked list of returned IPs. You can see this behavior yourself if you attempt to telnet to google.com, for example. You will notice that it takes a long time to finally fail. That is because google.com has a lot of IP addresses, and the telnet client was trying every one.
While round robin DNS does not typically have feedback into the status of the servers it is providing addresses for, it may help if you then have some sort of load balancer (including router-based tricks) for each of those addresses.
There are tricks to update DNS as things fail; if this happens, round-robin DNS with suitably short TTLs can be a pseudo-load balancer.