A lot of jokes ITT. Understandable with all the limelight they have been in lately. I'll try to give you a straightforward answer based off of my experiences. I joined in 2007, went to bootcamp in June of 2008, and reported to BUDS around December 15th 2008. I DORd (I quit/rang the bell) the day after Easter in 2009 on the first day of hell week.
It's about as easy as joining the Navy is for everybody else. There are certain minimum requirements you have to meet like most rates: minimum ASVAB, certain medical conditions are disqualifiers, no drugs (can be waived), no criminal background (maybe waivable, Im not sure), a clearance, and pass the PST (physical screening test). Basically, if you meet the requirements and there is space available, then you sign the contract and wait to ship out.
When I joined in 2008, they were really pushing for more guys to go to BUDS. They were hoping more guys going would equal more guys becoming SEALs. It doesn't work that way. Anyways, the class sizes were larger than normal, and some guys may have got in with waivers that wouldn't be accepted now. The classes also seemed to have more Fleet Returnees than normal (guys from the regular Navy who had packages approved to go to BUDS).
Once you get through boot, you go to BUDS prep. It is a pretty chill school. There are a couple active duty guys who run the school house, but most of the training is run by civilian coaches trying to get you ready for BUDS. It is a pretty relaxed environment comparatively. At the end of this training you must pass another PST with jacked up requirements, but still manageable. If you pass, you're on to Coronado where the actual training begins.
That's just how you get there. BUDS is where the process really begins. All the preliminary stuff isn't too difficult or too selective, although a few guys get lost along the way. Sometimes medical stuff comes up or they get into trouble. A few also get intimidated by BUDS before they even get there and choose to stay in the regular Navy.