I'm sure that you will get a lot of answers on this, but the way that we approach a trip seems to be consistent with most other sailors.
As boasun indicates, you establish a series of waypoints that you would like to achieve. You can enter them on a paper chart, a GPS, or a chart plotter. Regardless of the media, it's the same process. Of course, no plan survives action (von Moltke), so when you get offshore, you may find that you have to deviate from your intended course. In fact, you can virtually count on it due to current, wind direction, leeway, etc. Keeping track of those deviations is what DR is all about.
For instance, you intend to sail 000 deg, but can only make 040 because you're going upwind. As soon as you make the change from your intended path, you note the new course, speed, and time on the chart. At some point in the future, you advance your position on the chart and that is where you should be, but probably aren't. The other factors affecting your actual course are wind, leeway, etc. This is where the "art" in DR comes in. The better you know how your vessel acts in various conditions, the better you will be able to estimate the other factors. This is where boasun indicates that fixes are used. Near land, you can plot fixes and running fixes from known reference points on land; offshore, it's more of a challenge and usually relies on celestial sights.
IMO, DR is becoming a lost skill due to the ubiquitous GPS In many cases, we ignore all the external factors pushing us off course and simply plot a new course to the waypoint. For instance, on Victoria, we have a laptop running chart software, a handheld Garmin, and a iPod Touch with Navionics software. And I still DR just to keep up skills because a) the laptop can freeze, b) the Garmin batteries can die and c) the iPod isn't waterproof. IMO, DR is an essential skill that's being ignored. While at University, I took the Navy ROTC navigation course (DR and celestial) and have benefitted immensely. The CG Auxiliary offers similar courses (my uncle taught one). Every boater should have a working knowledge of DR.
True story: 30 years ago, just after I became engaged to Victoria, we were sailing up a channel in Ocean City, NJ at night. We had a beam wind and an outgoing tide so I was offsetting course to accommodate the conditions. The buoys were unlit and I couldn't see them due to background land light. Halfway up the channel, there was a loud crash and a nun buoy came rolling down the starboard side. Vicky and I still disagree on whether the collision was superb DR (my view) or idiocy (her view).
Another story: When I was 19, I DR'd the family Cat 30 from Ocean City offshore to Block Island with 2 friends. We noted each significant course change on the chart. When the wind veered to the west, I worried that we'd be offset east, missing Block Island. So we headed more northerly (noting time, course, speed), intending to intercept Long Island near Montauk. The plan worked and we turned east at LI, eventually arriving at BI without incident. DR is simply the techniques associated with position estimation.