This is another attempt to explain a condensed matter physics concept in comparatively nontechnical language. Comments are, as always, appreciated.
One common example of a quasiparticle is the polaron. When a charge carrier (an electron or hole) is placed into a solid, the surrounding ions can interact with it (e.g., positive ions will be slightly attracted to a negatively charged carrier). The ions can adjust their positions slightly, balancing their interactions with the charge carrier and the forces that hold the ions in their regular places. This adjustment of positions leads to a polarization locally centered on the charge carrier. The combo of the carrier + the surrounding polarization is a polaron. There are "large" and "small" polarons, defined by whether or not the polarization cloud is much larger than the atomic spacing in the material. Polarons are a useful way of thinking about carriers in ionic crystals, materials with "soft" vibrational modes (such as the manganites), and organic semiconductors (very squishy, deformable systems held together by van der Waals rather than covalent bonding).
Not content to let the general relativity fans have all the fun, I can describe this with a ball-rolling-on-a-rubber-sheet analogy. The ball is the charge carrier; the deformation of the rubber sheet is the polarization "cloud". Consider tilting the rubber sheet - this is analogous to applying an electric field to the material. The ball will roll in response to the tilt, but it will be slowed down compared to how it would roll on a hard tilted surface, since it has to put energy into deforming the sheet. In real materials, this shows up as a correction to the effective mass of the charge carrier. All other things being equal, polarons are heavy compared to bare quasiparticles.
We can carry this analogy further. Suppose we have two balls on the rubber sheet. In this classical picture, if the balls are so close together that their sheet deformations touch, the balls will be attracted together and end up in one deformation, held apart by their mutual hard-core repulsion. This is a crude analogy for bipolaron formation, which does happen in real materials. (Though, in real bipolarons the (purely quantum mechanical) spins of the individual polarons are important to stabilizing the bipolaron. The spins form a singlet....) Furthermore, suppose the rubber sheet takes some time to respond to the balls, and takes some time to restore itself to its undeformed state once a ball passes by. You can picture a ball rolling in some direction, leaving behind itself a little groove in the sheet that "fills in" at some rate. This would lower the energy of some other ball if that other ball were traveling in, say, the exact opposite direction of the first ball. This is a very crude way of thinking about the attractive pairing interaction between electrons in low temperature superconductors.
Finally, suppose the rubber sheet is really stretchy. A ball dropped on the sheet will pull the sheet down so far that it'll look like a little punching bag. Now if you try to tilt the sheet, the sheet will have stretched so tightly that the ball won't want to roll at all. Instead, the little punching bag will hang there at an angle relative to the sheet. Something analogous to this can happen in real materials, too - polarons can self-trap. That is, the charge carrier deforms the local environment so much that it basically digs itself such a deep potential well that it can't move anymore. Chemists have their own name for this, by the way. A molecule that deforms to self-trap an extra electron is a radical anion, and a molecule that deforms to self-trap a hole is a radical cation.