Roughly, deliberate practice is a means to improve yourself by intelligently breaking your target into components that can be practiced and upping the level of difficulty of each practice session to be just outside your comfort zone.
Deliberate practice is often associated with the idea that even genius-level talents (Mozart, Tiger Woods) got there through practice, not talent. However, you can ignore the talent debate and just concentrate on the idea that if your talent level is X, deliberate practice is how you get to 10X.
There are two things that I think need more emphasis when people talk about deliberate practice.
Deliberate practice is more than 10,000 hours
Or maybe it's less than 10,000 hours. Almost everyone I've used the phrase "deliberate practice" with has come back saying, "oh, that's the 10,000 hour thing."
They're referring to research saying that after 10,000 hours of practice a person will have reached their improvement limit. This is the least useful thing to know about deliberate practice research. It's the fine print warning that says if you keep a deliberate practice regime up for ten years it's going to stop being effective. Big deal! Most of you won’t even get close to 10,000 hours.
The ten thousand hour framing obscures two things, that deliberate practice can be applied to much smaller things (not just when your goal is to be world class) and when you want to make a smaller commitment. What happens after 1 hour of deliberate practice? TONS!
Every hour spent practicing is time when you're improving. There's not 10,000 hours of work followed by a single leap. It's 10,000 individual hours paired with 10,000 individual gradations in improvement.
The first research paper that I read on this topic is an study of competitive swimmers at all levels, The Mundanity of Excellence. Even at the youngest levels, they found that the fastest swimmers practiced better.
So, you're an office worker who sends tons of email? Take one hour, read this article on writing effective emails, and then rewrite your last ten emails according to those guidelines. Forever after you'll be a better emailer.
The key word is deliberate
A lot of people practice. They put in hours of work hoping to get better. Generally a high volume of practice does lead to improvement. But that's not the key insight of deliberate practice.
In the swimming example above they found that there were many similarities between the faster and slower swimmers, including how much time they spent swimming. The difference is that the slower swimmer would spend practice thinking about the hot tub and the faster swimmers would spend practice working on some minutiae, like how slight variations in the cupping of their hand effected the efficiency of their swimming stroke.
I generally find that it's easier to work more than it is to work smarter. Why is that? It would obviously be much more efficient if my preference were reversed. For example, my number one productivity boost comes from keeping an obsessively updated todo list throughout the day. My natural inclination toward the todo list is to compete with myself to see how many items I can check off. The biggest problem with my todo list is that I'll put everything on it and I don't spend much time prioritizing. So at the end of the day my todo list reflects more activity than accomplishment.
In the language of deliberate practice, the "skill" I'm trying to improve is productivity. The naive approach is just to work harder. The deliberate approach is to break my productivity target down into smaller pieces and train up the areas where I am lacking.
When it comes to productivity, I'm not afraid of hard work or long hours. Those are positives (I think). My weaknesses  are that I don't like making plans (I distrust them), I often don't follow my own plans, I procrastinate whenever the next step is not something I'm interested in (I almost lost an entire week to a screencast that still hasn't happened).
One of my old weaknesses was losing track of what I was working on and getting sidetracked. I solved that weakness by adopting a todo list. I bet if I spent more time "practicing" productivity I'd come up with an even more nuanced view of my strengths and weaknesses.
So, how would I train my own productivity? Each of those weaknesses needs a training plan. I've never seen anyone break down "productivity" in the way a coach would break down a training schedule. Should I say next Thursday I'll re-prioritize three of my old todo lists, take a coffee break, then prioritize three more lists (written 2 x 3 x prioritize todo list; 2:00 rest)? That would be taking a small subset of my productivity goal and training it.
I generally like to try to include my deliberate practice as an organic part of the rest of my work. That means I like to practice while I'm working instead of creating artificial exercises like the todo list one above. I'm not advocating this as the most hardcore way to approach work, but it's as hardcore as I've managed so far. The way I broke down training productivity was just to create a meta list that gives me points for things like: making a plan, working to a plan, and not surfing random websites. I'd be interested in a more disciplined approach though--does anyone have any training ideas?
 The book The Cyclists Training Bible has cyclists identify limiters, factors that are holding them back from achieving their goals. Then the cyclist puts together a training program that specifically addresses these limiters. The difference between weaknesses and limiters may seem subtle, but I think limiters are a much more functional way of looking at your weaknesses. I'm a terrible singer, but that hasn't held me back from anything meaningful. I'm also terrible at visual design and that often slows down my work. Gee, which one should I work on? Thinking about limiters also lets you work on things that you're good at but which happen to be extra important to your goals. For example, as a programmer I'm a reasonably good communicator but I've still managed to collect a huge list of regrettable programming outcomes that could have been solved by earlier, more articulate communication. So communication is always one of the skills that I'm working on.