The forces you feel in a moving car—those that push you back into your seat when the driver steps on the gas or throw you side to side when the car makes sharp turns—are everyday examples of fictitious forces. In general, these influences arise for no reason other than that the natural frame of reference for a given situation is itself accelerating.
The term "fictitious force" has a precise meaning within Newtonian mechanics—in fact, it's always proportional to the mass of the object on which it acts.
An elegant example of these types of apparent influences is the fictitious Coriolis force, which is responsible for the stately precession (or circular rotation) of a carefully suspended pendulum's plane of swing. If such a pendulum were suspended directly above the North Pole, it would appear to rotate 360 degrees every day. If you viewed this pendulum from a stationary point in outer space, however, it would appear to swing in a single, fixed plane while the Earth turned under it. From the outer space perspective, there is no sideways force (that is, perpendicular to the plane of swing) deflecting the sway of the pendulum. That is why the somewhat pejorative term "fictitious" is attached to this force. Likewise in the car, there simply is no real force pushing you back into your seat, your senses notwithstanding.
Nevertheless, analyzing a situation in terms of fictitious forces may, in fact, be the most effective way to understand what is actually going on. Take a stirred cup of tea, a charming example of a consequence of the Coriolis force. If a few tea leaves are present in the cup, they end up in a pile at the center of the bottom surface (and not along the edge, as one might expect, as a result of the also fictitious centrifugal force). If you imagine yourself rotating around in sync with the stirred fluid, most of the fluid would appear to be at rest while the cup would be counter-rotating around you. That rotating cup drags some adjacent fluid along with it. Meanwhile, near the bottom, the Coriolis force on that dragged fluid pushes it toward the center of the cup, carrying the leaves along with it.
With general relativity, Einstein managed to blur forever the distinction between real and fictitious forces. General relativity is his theory of gravity, and gravity is certainly the paradigmatic example of a "real" force. The cornerstone of Einstein's theory, however, is the proposition that gravity is itself a fictitious force (or, rather, that it is indistinguishable from a fictitious force). Now, some 90 years later, we have innumerable and daily confirmations that his theory appears to be correct.