1. You pay a monthly fee to download as many songs as you like to your computer. You may have to pay a higher fee to be able to be able to be able to put it on a device. There is usually a limit to how many players you can transfer to. I would hope that the Zune Marketplace includes the ability to transfer this music in the price of the subscription. I'm not sure of all of the technical aspects of this, but I believe this is how it works for most services: All of the songs you download using the subscription will have a license attached to it which allows you to play the songs.(I'm hoping you know a little bit about drm - digital rights management) You will not be able to burn any of these songs to cd. This license has an expiration date. If you keep paying, you keep the license current and you can keep playing the songs on your computer and on the player. You stop paying, then they will no longer work. I am not sure if they delete themselves or not.
You actually do not own the music. You are merely renting it. This is a disadvantage, but it does allow you to download thousands of songs for a relatively small monthly fee rather than pay thousands to buy songs that you may not like a month or a year from now.
If you like a particular song or songs enough, you can purchase them individually. I'm not sure how the Zune Marketplace works, since I do not own a Zune nor wish to own one. Most, if not all services allow you to burn the songs that you purchased.
2. Subscribe to own. The only services I know of are eMusic and Audio Lunchbox. You pay a monthly subscription fee and you can dowload a set number of songs. Different levels have different download limits. With these two services you actually own the songs and they are drm free. They are unprotected mp3s. You can transfer them to any device, including the Zune and iPod because they are mp3s and you can burn them to cd.
These two services tend to skew toward more independent artists. You'll find some mainstream stuff, but if you're really into Britney Spears, you're out of luck. Another limitation is that you're not going to be able to download thousands of songs in a short period of time. It could take many months or years depending on your level of service. I think they come with a free trial so that you can try it out and see if it's to your liking.
I like the idea of subcription services - both models that I mentioned here. I don't really need it since I have 1000 cd collection. I have an iPod, so I can only use the second one. The Zune can only use the Zune Marketplace. If you have or are interested in any other device, look for the "Plays For Sure - subcription" logo on the box. That way, you know your device supports subcription services as well as downloads. There are some players that are "Plays For Sure" but only support purchased non-subscription content.
Sorry for the long post. Hope this helps you.