Apologies to William Shakespeare for that terrible pun in the title, but I couldn't resist.
When the time comes to address your wedding invitations, you basically have three options: computer-generated, handwritten, or professional calligraphy. Your invitation sets the tone for your wedding. Only you can decide what you want that tone to be.
I don't recommend computer-generated envelope addressing if you're doing it yourself. Thanks to the near impossibility of getting your printer to accurately print onto specialty envelopes, it requires the use of stick-on labels. Those are fine for holiday cards and children's birthday parties, not for your wedding. If your stationer offers the option of pre-addressing the invitation envelopes (much like they print the return address on the envelope flap and the mailing address on the response card envelope), that can be an easy route to elegance. But be prepared to pay for it.
Hand-writing the mailing addresses on your wedding invitations lends a nice personal touch (and has the added advantage of being extremely low-cost), if you have nice penmanship or know someone who does. Depending on the size of your guest list though, this could be a daunting, time-consuming task. Unless you are truly in a budget crunch, I don't recommend it if you're sending more than 50 invitations.
The most elegant option, of course, is to hire a professional calligrapher. When a large envelope ornately addressed arrives in a mailbox, the recipient knows it contains something special. Calligraphy can be rather expensive, but if you have room in your budget and want to give your invitations that something special, professional calligraphy is the way to go. Calligraphers often charge by the line, so if your guest list includes a lot of apartment-dwellers, be aware that this will increase the total cost.
If you would like to feature calligraphy as part of your wedding but can't afford to have the invitations addressed, you can have your escort cards calligraphed, or have one master menu calligraphed and then reproduced by a local print shop. And if you decide to have the invitations calligraphed, have one addressed to yourself as a keepsake and also so your photographer can capture it when he/she photographs your stationery suite.
Your wedding day is likely to be the fanciest day of your life. It's only fitting that the writing should be fancy too!