The first thing people usually do when they get their telescope is to get better eyepieces. There is normally one or two eyepieces delivered along with a telescope but they’re usually low quality Huygens or perhaps Kellner eyepieces. For example, my Mak 127 came with two Kellner (actually modified achromat) of 10mm and 25mm focal length. Eyepieces differ by their construction, including the number of elements (lenses), their focal length, their field of view, their eye relief, and their field stop (which I won’t discuss here because this property is only interesting to more advanced astronomers). The current standard is the 4-element Plössl eyepiece, and its variants, usually called Super Plössl.
Once you settle for a decent quality (Plössl or better), the main property of the eyepiece really is its focal length. Basically, combined with the focal length of the telescope, it determines magnification, or power :
Magnification = telescope focal length / eyepiece focal length
A low focal length means high power, and a high focal length means low power. This is important because there is a limit to magnification. The rule of thumb to estimate that limit is to double the aperture (in mm). For example, my Mak 127 has a 127mm aperture, placing maximum magnification at around x254 (actually, it can go as high as x300 if seeing – atmospheric conditions – is excellent). I didn’t take that into account when I bought my first Plössl eyepieces, which were apparently meant to be used with a low focal ratio Newtonian telescope, and included a 4mm eyepiece. Well, 1500 / 4 gives x375, well above the magnification limit.
Another property to consider is eye relief. Basically, a longer eye relief makes you able to see the image at longer distances from the eyepiece. Plössl eyepieces have short eye relief by design, but this is more severe at short focal lengths since eye relief is a function of the focal length (for example 70-80% in Plössl eyepieces). One way to work around the problem is to get eyepieces of medium to long focal length (e.g. 12mm or more) and use a x2 Barlow lens, an element that comes with any new telescope or eyepiece set and doubles the effective focal length of the telescope. Another is to get eyepieces of a different design with longer eye relief, which I did. I can now comfortably look at Jupiter with a 6mm Super Plössl eyepiece (close to maximum magnification) and enjoy the excellent sharpness.
Plössl eyepieces usually have 50° of apparent field of view (aFOV). Some eyepieces have a wider FOV. For example, I have a 66° WA (Wide Angle) 15mm eyepiece which I can use instead of a 32mm to locate objects at a higher magnification.
And last but not least, check the barrel diameter of the eyepieces. The most common size is 37.5mm but there are accessories with 50.8mm barrels.
See Tele Vue Optics: Advice for more details about choosing eyepieces.