Chances are that your apps are not OS X native. Check the readme or sites of the apps to see if they are. However, not to fear, non-native OS X apps can launch under Classic Mode in Mac OS X. The only downside to using Classic is that it can be slow, and you need LOTS of RAM (384 MB or more) to do it efficiently.
There are three different possibilities each of your applications may fit under.
1. Classic Application: These will run only in Classic mode. Now in OS X, it's pretty seemless, but there are some apps that just don't like it. Thankfully this is few and far between.
You can tell if the application is ONLY for classic if, when you double-click on it, the menubar goes back to the OS 9 platinum look & feel. As well, it will launch Classic mode if that is currently not running.
If you want an OS X version, you'll need to either upgrade or re-download the OS X-ified version.
2. Carbon Application: These will run in OS X without the need of Classic mode running. You can tell if an application is "carbonized" if it runs in OS X as well as OS 9 without changing the menubar in OS X to the platinum OS 9 one, or boots up OS 9 if it isn't already active. For example, pretty much every Adobe application is Carbonized. One installation and you can run it either in OS 9 or OS X. (PageMaker seems to be the only one not carbonized from what I've seen).
Carbonized applications contain all the Classic elements inside it as well as all the necessary aspects needed to run in OS X. You will probably not need to upgrade or download a new version unless one comes out and you decide you want to.
3. Cocoa Application: These applications ONLY run in OS X. Chances are slim you have any at all, if indeed you haven't installed OS X and until now are really just running Mac OS 9.
When in OS 9, you can tell an application is a Cocoa application because it usually shows up with the file extension ".app" at the end of the file name.
All the applications that come pre-installed with OS X (Mail, TextEdit, Preview, etc.) are all OS X Cocoa applications.
Cocoa applications contain no OS 9 code.
If you are still just running OS 9, your applications will all fall into the first two categories.
All your applications *should* work in OS X. Classic mode, for the most part, is pretty damn good. But the only realistic way to be sure is to try each application you wish to make sure is going to work while running OS X.
All the OS X pre-installed applications work fine for people to use, so alot of the OS 9 variants will be unnecessary and, providing you never boot back into OS 9, will never be used.
I do suggest keeping them though. OS X may be the most stable OS on the planet, but nothing is perfect. RtC can attest to the wackiness that can ensue when you do get hit with a seemingly rare or obscure bug/problem that might force you to jump back to OS 9, but for probably 99% of OS X switchers, you will only ever see OS 9 for "Classic Only" applications.
What works in OS X?
Well, all the key applications in OS 9's Applications folder are represented with Carbonized or, more likely, Cocoa variants. You may have to set up some importing of things (like bookmarks from your classic version of IE) and obviously network settings and personal setting information will need to be done. But most likely the only software you will need to change will be third-party programs.
Some of these are available as a free download and others will require a new purchase of an upgrade kit.
As for shareware/freeware ones. I'm willing to bet that many will be carbonized right now and will simply require you to run them to verify this.
Any that are, I also recommend moving to your OS X Applications folder. Or at least a copy of it for ease of finding.
Installing OS X:
This is something alot of people wonder about. *MY* suggestion is to partition your hard drive into two or three (or more, if you feel like it) sections. Have one for OS 9 that is 1GB or less (I think 500MB would also be fine. I use 1GB so there's some breathing room)
Install OS X on a partition of at LEAST 5 GB. Anything smaller and you may find you need more room (It's actually not that big, but for some reason, I found some of the larger updates ran out of "swap space" in a 4GB drive about 6 months later...)
I also have at least 1 more partition where I like to store all my files and such. If you ever need to re-install your OS for any reason, you can store a back-up of your Home directory on this partition and move it back later.
A nifty trick I use for Music/MP3s:
I also have a separate partition for my MP3s and I have turned my "Music" folder into an alias of my MP3 folder (Empty out the Music folder, if necessary, delete it, make an alias (command-L) and rename the alias "Music" and put it back where you deleted the older folder).
Your iTunes data will migrate to the other partition and, if you reinstall your OS, you keep your playlists intact if you forget to back up. As well, iTunes *WANTS* to put all your MP3s in ne folder and that just uses up your OS X drive.
This doesn't work for the OS X Applications folder or for the general Users folder, from my experience.
The "Documents" folder is another popular one to do this with.
Hoepfully I haven't confused you further. I just realized how much I've rambled on...
Perhaps I should write a book! "Strongblade's Guide to OS X!". I could even include a spiffy picture of myself on the back cover...