A telescope is no good without a good quality mount and neither are any good to you without some good quality eyepieces. The function of an eyepiece is to magnify the incoming rays of light and present them to the eye of the observer. Choosing which type of eyepiece to buy from the 6 or so popular designs can be bewildering but there are a couple of things to consider that will help.
If you have a ‘fast’ telescope (a telescope with a low f/ratio such as f/4 or f/5) then they tend to be less forgiving and higher quality eyepieces should be aimed for whereas ‘slower’ telescopes (with a high f/ratio around f/8 or f/9) can cope with lesser quality eyepieces and still yield a good sharp image.
Something else to think about is eye relief. This is the distance between the outside of the eyepiece (or more accurately the outer surface of the lens) to your eye. Over the years I have tried many different eyepieces, some with such a low eye relief that I really had to squash my eye up to the eyepiece to see the image properly. Its all down to comfort and ensuring you see the full field of view rather than just the central part of it but its of more concern to those who wear glasses to observe. If this is you, then you should look for an eye relief of about 20 mm.
The real challenge though when choosing eyepieces is which focal length or apparent field to go for. Like telescopes, eyepieces have a focal length and this determines the magnification that you will get with your particular telescope. Its a simply calculation; divide the focal length of the telescope by the focal length of the eyepiece to work out the magnification. For example, a telescope with a 1000mm focal length and a 15mm eyepiece will give a magnification of 66x so you will see an image 66x bigger than with the naked eye.
As you observe, you will realise that you need more than one eyepiece. You will get much more use out of the middle range of magnifications so concentrate on getting these first, perhaps a 25mm and 9mm eyepiece. You can then look to purchase a lower and higher focal length at a later date. Ultra stable atmospheric conditions for the use of a high power eyepiece are few and far between so there is no rush to get one of these, you might use it for just a few nights of the year.
Another eyepiece accessory you could look to buy is a barlow lens. These fit in between the eyepiece and telescope and act to either double or triple the magnification of that eyepiece. Careful planning of your eyepieces and a good quality barlow lens will double the range of eyepieces. For example, there is no point buying a 20mm and 10mm eyepiece with a 2x barlow lens as it will make the 20mm eyepiece work like the 10mm.
You can use the calculation above in reverse to workout which eyepiece you need for a particular magnification. Simply divide the telescope focal length by the magnification you are after and that will tell you the focal length eyepiece you need.
Field of View
The final note about eyepieces is field of view. There are two terms here, apparent field of view and true field of view. The apparent field of view is the width in degrees of sky as seen if you look through the eyepiece alone. Not much use really but using this figure, along with the magnification your eyepiece/telescope combination gives (from above) and you can work out the true field of view. This is the actual portion of sky you will see when looking through your telescope. To calculate this, divide the apparent field by the magnification. For example, if you have an eyepiece with an apparent field of 50 degrees and a magnification of 100x then you will get a true field of view of 0.5 degree which is just about the size of the full moon in the sky.
There are other types of eyepiece out there, for example those with illuminated reticules inside. These have either fine wires inside the lens or lines etched onto the glass that are illuminated by a faint red light. Through the eyepiece you will be able to see a gently glowing red cross (or in other designs angular measurement scales) which are of great use in astronomical imaging and other more advanced applications.