Adding to Barts reply:
My best example of aliasing is rotating wagon wheels in western movies. That is temporal aliasing, but the principle extends to spatial aliasing that we are concerned with here. The film is shot at 24 frames per second. If the wheel is rotating at 24 times/second, it may seem to "stand still" because each movie frame capture a wheel that seems identical to the other*). If the wheel is rotating at slightly more that 24 times/second, it might appear to move backwards. Clearly, this is a "false" impression, caused by an imaging system that is not able to capture the true scene, and instead captures/renders a untrue scene. If the camera had had a long exposure time ("aa filter"), it would smear the wheel into a fuzzy thing that may or may not appear more realistic - at least there would be less room for confusion as to what really went on.
There will always be aliasing in your image (excepting some corner cases that are probably less interesting for photographers).
An AA-filter will only _reduce_ aliasing, not completely remove it. It will do so at the cost of some real image detail. Those details can to some degree be brought back by sharpening/deconvolution. If the product of scene detail, scene/camera motion, lens/diffraction blur etc is sufficient to suppress aliasing to the point where it does no harm, adding an aa-filter may be unwanted, as it will still cause a slight attenuation of image sharpness.
The visibility and degree of annoyance depends on the scene, the viewer, the print and recording technique. Scenes with highly periodic fine detail (e.g. feathers) are known to cause more annoyance than scenes with less periodic fine detail (e.g. leaves).
The use of Bayer color filtering (and debayer in raw development) makes the analysis more complex, but the basic principle is true even for Foveon-type sensors (contrary to popular internet belief).
-h
*)Assuming that the camera has a very short shutter time (generally untrue) and that the wheel has no rotational symmetry (generally not true).