Bear in mind that I was not directly involved in the writing of the Agreement (the law firm did that) and this is my own personal explanation and not anything official (I'm just tired of repeating myself to worried people).

Preamble
The preamble simply defines what and who is being discussed and makes a statement that the terms of the submission agreement apply just to the use of the materials on the deviantART website.

1. Term
Thi simply explains that the agreement is in effect from the time that you submit something to the time it is removed either by you or by deviantART. As far as the 'notification' portion is concerned a simple statement made to the Helpdesk will probably suffice, although we do keep records of what gets deleted and when so notifying us is probably not as big of a deal as some people might make it out to be.

2. Ownership
Simple- you own your art at all times and can continue to do whatever you want with it.

3. License to Use Artists Materials

3a.
This gives deviantART permission to encode your work so that it can be compatible with different viewing formats. It allows for future technological advancements and new formats which we may want to use in the future.
3b.
This gives deviantART permission to do whatever it needs to do to show your work to a viewer. It also allows us to take advantage of new technology which could help us show your work to a wider audience.
3c.
This gives deviantART the ability to alter your work in whatever means is neccessary to display it.
3d.
This gives deviantART the ability to 'sublicense' someone else. This is important when deviantART advertises or has interviews where $spyed (for example) wants to pimp your stuff like he did here [link] on The Screen Savers. It also comes into play with our Pasties system.
3e.
This means that if your work is used in an interview like [link] you can't claim any right to the show.
3f.
This just gives deviantART the right to use your work in advertising, although judging by past behavior you'll probably be approached first to make certain your okay with that.
3g.
This just defines the term 'Artist Materials' as being anything which you submit to us and your biographical information (for use in interviews and advertisments and such).

4a/b/c. Name and Likeness
This simply states that deviantART can use your biographical information and likeness (if available) in promotions and advertising.
It also states that you, and anyone who is featured or named anywhere in your submissions, won't get angry about it.

5. Limitation Of Rights
Basicly everyone is always freaking out about how they believe that the Agreement allows deviantART to sell their work without permission. This clause states that deviantART has to notify you and must obtain your consent before your work goes into anyone else's hands for any commercial reason.

6. Payment
This states that if you submit something to deviantART, deviantART does not owe you any money just because you put something on the server.

7. Representations and Warranties
Basicly a-h state that you are legally able to enter this agreement, that you have the legal ability to submit the material which you upload, that your submissions won't violate someone else's rights or break any laws and they don't contain harmful or malicious code (like viruses).

8. Third Party Payments
This means if you need to buy materials (like stock photographs) for use in your submission you pay for them, not deviantART.

9. Indemnity
This essentially says that if your submissions or behavior bring a lawsuit (or something similar) against deviantART you agree to take the full responsibility and take whatever costs and penalties on yourself.

10. deviantART's Performance
This states if the site breaks or a server bursts into flame you can't complain or expect any refund (for subscription time for example).
It states that we can discontinue any third party arrangements at any time and that deviantART itself could dissappear and never return for whatever reason and without notice.
It also states that we are not responsible for what you submit here if it gets deleted, removed, lost, or corrupted and that you need to keep your own personal backup copies and that deviantART is not going to keep a close eye on your content for you.

11. No Warranty
This simply states that deviantART does not have to be available every second of every day and may experience downtime, that you are not guarenteed sales through your Prints and Shop items and that deviantART is not guaranteeing anything else (like lots of pageviews or instant popularity and such).
It also states that we aren't responsible for any viruses or other nastiness that somebody might sneak onto the site.

12. No Liability for Third Party Use
This essentially says that if some jerk steals your art, it's not deviantART's fault that it happened.

13. Exclusion of Damages
This is a standard "If Using It Hurts You Or Breaks Stuff It's Not Our Fault" disclaimer that you find plastered everywhere nowadays.

14. Copyright and Trademark Notices
Obvious.

15. Miscellaneous
15a Governing Law
This states that regardless of where in the world you are the Law of Los Angeles California is what applies, not your local laws.
15b Assignment
This states that deviantART may assign (or turn over) the Agreement, or parts of it, to somebody else- such as a representative or a buyer, etc. I assume the only thing that changes when this happens is that you'd be dealing with a different party instead of deviantART.
15c Notice.
This tells you where and how to mail stuff to deviantART.
15d Entire Agreement
This states that this Agreement overides and replaces all previous versions.
15e Modification
This states that deviantART can change this agreement and that it will notify you if and when it does this.
15f Survival
This means that certain parts of this agreement continue even after it gets terminated-
The Term (1) stays the same of course.

License 3e remains- this means if your work was used in an interview like this [link] you still can't claim any right to parts of the show.

Name & Likeness 4b remain in effect which means that you and those real people shown or named in your old submissions still can't get mad at having been on deviantART.

5. Limitation Of Rights stays in effect which means your work is still 'safe'.

6. Payment stays in effect which means deviantART still doesn't owe you money because you submitted to it.

7. Representations and Warranties stays in effect- that's all the stuff about your submissions having been legal and not breaking any laws and such.

8. Third Party Payments remains in effect which means your debts are still yours, not deviantART's.

10. deviantART's Performance remains which means deviantART is still not guaranteeing a bug-free and break-down free existance.

11. No Warranty sticks around which means deviantART still isn't guaranteeing anything.

12. No Liability for Third Party Use is still active so you still can't blame deviantART if somebody stole your stuff.

13. Exclusion of Damages survives so if you got hurt or broke your stuff while using deviantART you still can't blame the site.

14. Copyright and Trademark Notices is still around so all the stuff that was copyrighted or trademarked is still copyrighted or trademarked.

15. Miscellaneous doesn't go away either which is this stuff you just now finished reading (if you made it this far)