For 3D printing, potential problems can run the gambit; your print can begin warping off the build platform, you could run out of material midprint, your material can tangle, clog or be ground to bits in the extruder. Solutions vary on the slicer and printer options so this is a general guide.
-Clogs can occur for a couple of reasons; if the heating element isn't hot enough, it will let the filament melt but then extrude poorly. If your gears pulling the filament are worn down or have filament in between the grooves, it will move the filament slowly enough to clog the extruder. Flexible filaments can be extremely nitpicky and if your gear isn't maintained or is worn down, then its possible (and painful to find!) that the flexible filament has wrapped itself around the gear and your extruder is not actually clogged. On that note, I have used Ninjaflex before and then Woodfill after, letting it cool after the latter print. That created a very difficult to remove clog thanks to the nature of both materials.
-Possible solutions include feeding and removing sections of filament through the extruder to clear the clogged material. Some companies actually sell "cleaning" filaments, wherein they won't melt after feeding them into the extruder, like a pipe cleaner. In extreme cases, you can actually take a blowtorch to the extruder (remove the insulation and wires and everything else
First Layer Problems
-If the first layer isn't sticking to the print bed, you may want to consider recalibrating or swapping the bed material. Painters tape is a easy to use material and easily replaceable but does not have the same sticky properties of a glue stick, PET tape or hairspray. ABS prints stick well to ABS slurry (acetone + ABS) but can be difficult to remove or messy. Generally poor calibration is a main cause of first layer woes; too close and the material comes out smushed, not at all or the extruder tip can damage the bed. Too far and you will produce spaghetti or no filament will consistently come out.
-Sometimes filament prints can come out "hairy" thanks to the material coming out of the extruder. This can seem similar to stringing, wherein thicker strands of material follow the extruder movement. The primary difference is that these hairs are extremely small and thin. It seems so far that temperature is a prime factor in hairness, in addition to the type of material. For example, a print done in Ninjaflex has a higher possibility of being hairy; it prints at 1/3 of the normal speed of other materials and hotter. These two elements causes the extruder to just barely ooze, in turn, creating hairs. PLA does not have this issue nearly as much, but can still have it from time to time. This is fairly easy to post process, you just trim or burn off the hairs.
-Homing refers to to the extruder trying to find its original home position. If your printer is having issues finding it's home, you may want to check the touch switches at the end of rods. If the extruder grinds and tries to pull itself past its origin (meaning, its not stopping where it normally is,) you may want to check if the touch switch is connected or attached properly. I've had it happen before, where the touch switch was slightly ajar and wasn't detecting the extruder. The end result? The pulleys slipping thanks to them trying to pull something into position that was already there.
-If you find your prints learning or skewing to one direction or another, it may be the pulley system. If your pulleys are slipping, or are too tight due to friction or rubbing up against something, then it can cause leaning prints. Generally you will have to loosen a pulley assembly and move the pulley or replace it and reset it. For printers like the Makerbot 5th Gen or CubePro, you will need to contact their support teams, since this will require disassembly and possibly service by a member of their teams (you wouldn't want to void warranties!)
Out of material
-Running out of material during a print can be an easy mistake. Although slicers estimate the amount of material you'll need beforehand they aren't always precise, or you may accidentally release a print job without enough material. Generally you won't find out til its too late. Some printers will auto detect a lack of filament and stop immediately, others will warn you prior to running out. These are generally microchipped and keep track of the material, such as printers from the Cube family, but also the higher end Stratasys jetters.
Prints falling off
-Poor layer adhesion can attribute to part of a print falling off the rest of it. This can occur due to poor calibration. via layers barely being attached or the extruder being too close to the print. The latter can cause grinding or snapping of the part.
-An occasional occurrence to happen during prints is jamming, grinding or snapping of the filament spool. Although most spools are tightly wound, it is possible for a spool of filament to be wound underneath another section of the filament. When this happens, the filament does not feed properly can can be ground by the feed gear. The best solution is to make sure your materials are wound and taped or tied in place after use. Whenever you have tangled filament, be sure to remove it from the feed and then clean the gear with canned air or a clean toothbrush.
-A sign of underextrusion can be extremely thin lines of material on the print bed. Personally, if you feed filament in and it doesn't come out of the extruder neatly or smoothly, it can be a sign of a clog or underextrusion. If it occurs during printing, you may want to check the feed rate in your slicer.
-One of the major issues that can occur from the first layer onward is warping of the print. This is a common issue under certain conditions; if you are printing in ABS without a heated print chamber or build plate, the material will shrink and lift as it cools. Interlayer warping can occur due to lack of layer adhesion or lack of cooling/fan vents for PLA prints; if the PLA does not dry and cool fast enough each layer, it can peel off the layer below.