It's 6am EST, although people report often getting "in" a few minutes early, as early as 5:45am (which is the time the Disney website says, although when I last made ADRs in June, they didn't really open things up until 6am on the dot).
https://disneyworld.disney.go.com/en_CA/faq/dining-reservations/advance-reservation-restaurants/
I prefer to make ADRs on my desktop (on Firefox, although I don't think the browser makes much difference), but I always have my phone nearby to use as a backup in case of any glitches. I make sure I'm logged into MDE on both the computer and phone a few minutes ahead of time, with my list of desired ADRs, arranged in order of toughest to easiest to get, in front of me, and my search criteria for the first restaurant I want filled in and ready to go. If you're staying onsite and reserving for your length-of-stay, be sure that your hotel/ticket confirmation numbers are linked to MDE and displaying properly.
If you're nervous about it, I highly recommend practicing ahead of time. Simply pretend you want an ADR for 60 days out, and follow all the steps (stopping short of payment/confirmation) so you'll know what to expect.
*Another tip: sometimes, searching for a meal (e.g., "breakfast," "dinner") won't give you any results, whereas searching for a specific time (e.g., "8:00am," "5:30pm") will. I don't know why this is, but when in doubt, search for a specific time rather than a specific meal.
If at the end of the day, there's something you wanted and couldn't get, I recommend touringplans' free reservation finder tool, which will check back and notify you if something opens up (at which point, it's up to you to click through ASAP and try and snag the reservation before someone else does).
