Ask a dozen freelancers what business tools they use, and odds are you'll get a dozen different answers. Personally, I've used everything from just a text file, to spreadsheets, to just a paper notebook. Depending on the kind of work you do, any one of these may be sufficient. However, this method doesn't scale when your business grows past a certain point--meaning that one day, if things are going well, you're going to look at your system and and realize that you actually have no idea of if you sent invoices to certain people, or when you sent them, and whether they've paid, and whether you got in touch with Shirley last week.

It isn't necessary that you work out a system before becoming a freelancer, but at some point when you start making sales, it's good to start thinking about it before things get desperate and too disorganized. Depending on what type of work you're doing, issues you might need to track are:

• Assignments/projects - Including information such as what the assignment is, any specifics such as word count or course hours, how much it pays, and any terms on your contract such as "we pay within 30 days of receiving your invoice." Often, you will have to send an invoice
• Customers/clients - Including the company, the people you deal with their, their email addresses, phone numbers, IM contact information, and if you really have it together, any other personal information you have (birthdays, etc)
• Invoices - Including which ones you've generated because a project has finished, when they went out, if they've been paid, and what date payment is due
• Expenses and other issues - Anyone who is self-employed needs to become aware of tax issues for the self-employed in their particular country/state/city/etc. You're cheating yourself if you don't properly track expenses and so aren't able to write off as much as you should, and you're inviting an audit if you go overboard.
• Time spent - In the case of projects where you're billing per hour of work rather than per word, page, or project milestone. 

The important thing is that you find a system that works for you. I'm not a fan of telling people that they MUST do things a certain way. So instead, I'm going to help you to understand some of the software tools you have available. They generally fall into four classes, and unfortunately it appears to be difficult to find one tool that does all of this well for freelancers--if you happen to know of one, by the way, please mention it in the comments here and I'll look into it! These classes are:

• Project Management - Project management tools are more oriented around which projects you're working on. There's generally spots to define your customers, so you can specify who which project is for, but in many tools there's just a basic form for customer information. Often there's components such as Tasks (things to do), Milestones (things that need to be done by a certain date), and Messages/Notes (general stuff related to the project). Many of these tools also allow you to manage Resources (typically, people in the freelance world, though these can also be, say, a piece of machinery that can only be used by one person and project at a time). So, if you're working on a project with three people, you can define the three of you as Resources and then assign Tasks and Milestones to each of you accordingly.
• CRM - Customer Relationship Management tools are very popular among people in sales and other fields that are heavy on the personal contact front. These tools help you keep track of, depending on their level of sophistication, everything to do with a specific customer, such as company, specific people, notes on what you spoke about, reminders to contact or do something, and even the emails you exchanged. 
• Invoicing/Billing/Personal or Business Finance - Depending on your overall goals, you may choose to find a very simple program that lets you put in which customer owes you how much, generate the invoice, send it off, and then later mark it as paid or see in a report that someone's late. Or, you may choose to go a bit bigger and get financial software for managing a home/small business, which would get more detailed into items like Accounts Receivable, billings, expenses, charging taxes, and more. This software may or may not also integrate with income tax software. 
• Timekeepers - These programs are all about tracking how much time you spent doing what, for whom, at various levels of detail. If you're charging people by the hour, half hour, or any kind of time increment, it's important that you properly track this information. It's both possible to seriously overestimate and so overcharge your customer (and risk they'll go with someone else next time and may even have a bad taste in their mouth, which means no referrals or even actively telling people not to go with you), or you could undercharge and end up not making enough money to really cover your time, which can slowly bleed your business dry.
  

In later posts I'll talk about particular programs of interest. For now, the general idea is to decide which of these you need, and start finding software that runs on your particular operating system (Windows or Mac, typically). Find programs that you can try out for free and see if they suit your particular work style, or could suit you if you adjusted the way you used them. There are also Web sites that offer pay services where you can use their software from any computer. No matter which you choose, be sure to make regular backups. No matter what way you organize and store your business information--even if it's just on a pad of paper--losing it is way more painful than a little disorganization.