Sure thing, the first number (5) is the class of climb. A sidewalk would be 1; rock climbing begins (and ends) at 5.
The second number reflects the difficulty of the [hardest part of the] climb. I believe it originally went from 1-10, with 10 being the hardest thing a human could climb, but innovations in climbing shoes and better climbers have raised the max over the years, so now it goes from 1-15.
Imagine the scale as being of exponential difficulty. For the same climber who spent a week going from 5.5 to 5.6, it may take a year to go from 5.10 to 5.11. So you can imagine how elite a 5.14 climber is, let alone a 5.15 one.
The difficulty is rated as one pitch relative to another and certain pitches as benchmarks for a certain difficulty. e.g. pitch X is 5.13a and pitch Y is somewhat harder so it's 5.13c while Z is quite a bit harder making it a 5.14a, etc.
Grade. 5 means roped climbing, the decimal means how hard of a climb it is within the category of roped climbing.
5.6 can be climbed by any healthy adult, 5.7 by most. I reached 5.11 in a year of climbing; I expect to reach 5.12 within 5 years, and 5.13 sometime within my lifespan if I am dedicated and train relentlessly. 5.14 is only within reach of the greats who live their lives to climb, and 5.15 has been climbed by 2-3 people on this Earth.
The current grades are more a snapshot of history than the physical limits of humanity, IMO. Like the 4 minute mile, once one person did it, you saw several people after that able to do it. In the golden years of Yose, 5.12 was this crazy thing that only the elite did. Now it's something normal people can achieve. Part of that is technology -- I wouldn't want to ever lead 5.12 without a set of cams -- but part of it is that it's just easier to get to somewhere if you already know it's physically possible.
My experience is that most relatively fit folks can follow a face climb in the 5.7-5.8 range their first day out - some might manage 5.9 or 5.10. Again though, that's for face climbing which feels more natural and doesn't require much in the way of special technique. Climbing cracks (esp. wide cracks) is much more difficult for most people to progress on.
