How do I determine if I'm able to run a server?
Easy. Ask yourself the three questions of server running-ness ...? Yeah, lets go with that.

1. Do I know how to change the time in the server?
   
2. Do I know how to port-forward or use a REAL server instead of my cruddy computer?
   
3. Do I know what server modifications are available for usage?
   

If you answered no to any of these questions, save us all a little time and just join up either with someone who can answer yes to all these questions, or just go play with yourself.

I can run a server, but my friends can't connect to it!
This is a difficult question no matter how easy it sounds like it can be fixed.
There are numerous ways of failing:
[list=1]
[*]You are trying to make the end-user (your friend[s]) connect to a LAN address (e.g.: 5.0.0.1, 1.1.1.1, 192.168.0.5, 172.24.211.3, localhost)
[*]Your friend sucks at typing your IP address or hostname.
[*]Your server is whitelisted and you have forgotten to add them to the whitelist.
[*]You have not port-forwarded to your computer.
[*]You have not opened the port(s) for your server(s) in your firewall.
[/list]
... and many, many more. Those are the top 5 I've ran across though.

[i]I want to install a mod but I don't want to read the instructions.[/i]
Well that's your problem, not mine.

[i]What's the purpose of this article?[/i]
There truly is no purpose. Some people don't realize how technical these things can get, and this is a very, very low level of technicality. If you don't understand most of what has been posted here, you probably shouldn't be running a server.

The ideal situation for someone to run a server:
[list][*]A dedicated or virtual private server
[*]>2GB RAM
[*]~5GB free disk space (with a disk of faster r/w speeds)
[*]Bukkit experience
[*]Background in Java programming
[*]A brain
[/list]
Disclaimer: I do not mean to offend anyone, hurt anyones "feelings" or make them feel disappointed or inferior. This post is just to make you realize if you really know what you're doing.