There are three basic steps:

1. Copy the sermon onto your computer
To do this you will need:
A copy of the sermon on tape.
A tape player connected to your computer. Connect it using a lead going from the Line Out (or Headphone) socket on your tape player to the Line In socket on your computer's soundcard.
Sound editing software. If you don't already have this then you can download it as shareware (e.g. try GoldWave or Cool Edit).
Play the tape of the sermon onto your computer, using the sound editing software to record a digital audio copy of it onto your hard drive. You should save this audio file in a wave (.WAV) format. The size of this file depends on the quality settings you choose, but for a 20-minute talk we would expect a file of around 50Mb in size.

At this stage you can use the sound editing software to remove the parts of the tape that you don't want, such as the gaps before and after the sermon.

2. Convert the sermon to RealAudio or MP3
To do this you will need either of:
Software to generate RealAudio files. For this you will need a copy of RealSystem Producer, produced by RealNetworks. Don't pay lots of money, since there is a cut-down free version called RealSystem Producer Basic. You have to hunt around on their website to find it; last time we looked it was available here (but it keeps moving, so you may have to do a search).
Software to generate MP3 files. Your sound editing software may do this already, otherwise it is easy to find shareware which you can download.
Using this software, the .WAV file produced in the first step can be converted to a compressed format which can be streamed from a webserver to the user. The choice of whether you use RealAudio or MP3 to do this is up to you, but it is worth experimenting to see what works best. One advantage of using RealAudio is that it is easier for users to listen to it online; often with MP3 they will have to download the sermon before they can listen to it.

We recommend that you use a very high level of compression, sacrificing sound quality for a smaller file size. This has the advantage that only a small bandwidth is required for a user to listen to the talk online. Our experiments found that creating a RealAudio file with a bandwidth of 8.5kbps produced acceptable results (giving a file size of around 1.4Mb for a 20-minute talk). Using a higher bandwidth, such as 16kbps, gives much better sound quality but makes the resulting file correspondingly larger; this may cause you to run out of space on your website if you are hoping to upload even a moderate number of talks.

Having produced the compressed audio file ready for the website, you may also want to create a backup of the sermon in a high-quality MP3 format (using a bandwith of 64kbps gives a file size of around 10Mb for a 20-minute talk). This means that you can then delete the large .WAV file safe in the knowledge that you've still got a high-quality copy in case you ever need to recreate the file for the website.

3. Upload the RealAudio/MP3 file to your webserver
Contrary to what you may have heard, you don't need a special (i.e. expensive) webserver to play streaming audio files. Simply use your regular webserver and you shouldn't have any problems. Some of the functionality isn't quite as good as you would get if you spent lots of money, but it works fine for online sermons.

If you are using RealAudio then your sound file will have a .RM extension. Before you upload it to your webserver you need to also create a .RAM file which points to the .RM audio file. The .RAM file is made by creating a one-line text file containing the location of the .RM file on your website (e.g. http://www.mychurch.co.uk/sermon.rm) and then saving this text file with a .RAM extension (e.g. save it as sermon.ram). Then upload these two files to your webserver and provide a link to the .RAM file from a webpage. Users clicking on this link will be able to listen to the sermon online! (If you want users to be able to download the sermon then on the webpage link directly to the .RM file.)

If you are using MP3 then before you upload the .MP3 file to your webserver you need to create a .M3U file which points to the .MP3 audio file. The .M3U file is made like the .RAM file — create a one-line text file containing the location of the .MP3 file on your website (e.g. http://www.mychurch.co.uk/sermon.mp3) and then save this text file with a .M3U extension (e.g. save it as sermon.m3u). Then upload these two files to your webserver and provide a link to the .M3U file from a webpage. Users clicking on this link will be able to listen to the sermon online! (If you want users to be able to download the sermon then on the webpage link directly to the .MP3 file.)

Further comments
It takes a bit of fiddling around to get the first few sermons available online, but it gets much easier once you've got the hang of it! One church which is using the method described on this page is St Luke's, Wimbledon Park — take a look at their sermon archive to see how it works in practice.

If you want to read more about RealAudio then the WebDeveloper.com Secret Guide to RealAudio is helpful.