The disk will be dropped from the array and the work will continue as usual in all cases (for single disk read error / failure).
For multiple disk failures (when there's only one disk left on RAID1 array), you'll get read error but the disk won't be drpped. So... in most cases server will still work (if error won't hit system files area).
Eg. for mysql (If you only have databases on SSD which is best) you'll get "connection lost" error when reading affected row, and the work will continue. Same for apache in prefork - it'll fail to serve single file (it'll be much worse if the error hit some script that is included sitewide).
you will have to try to check the disk ( fsck )
then you try to rebuild ( mdadm )
if any of the cases fail ( check the disk first ) , you send a ticket for a replacement ,after the reboot it should rebuild in most cases.
You can't do fsck on live filesystem or on an unmounted raid mirror. That'll have no purpose anyway. You just use mdadm to add the drive back to the array.
You can run "badblocks" or "smartctl" (short/long test) on the failed drive to identify the errors (if any)... before adding it to the raid back so it won't affect the site performance.
For OVH techs...
* i think the replace time is awesome (for me about 10-15 minutes). You should give MODEL, SERIAL NUMBER, ID (/dev/sdX), ERRORS (smart, kern, badblocks, etc.). Just run smartctl --all /dev/sdxX and copy/paste.
* They of course won't replace the drive by themselves (it's impossible to tell if drive is damaged, it could be good even when there were some read/write errors before). So you must file a ticket (not ordinary one, but the other one, that is used to report damaged components).
* The bad thing is i think they'll just "pull off the plug". So you get damaged files and broken database table on your "good" disk. Then fsck will be done, and you'll be facing very painfull process of database check / recovery (if you have very big myISAM tables and you probably should rather be using myisam than innoDB when storing data on SSD).
That's really strange as techs have SSH access, can login and shutdown the server. OVH have recently developed SMS sending service so if they can't shutdown by hand they could just send SMS telling (shutdown the server in 1 minute). So i think, better to shut down everything before replacement.