There are several free options for designing logos that go beyond simple paintbrush programs. The best option for the job depends on conditions such as how unique the logo needs to be and whether it will be used on a website, business card or storefront sign.
The most simple way to make a logo for free is to use a tool such as LogoCreator, which lets users enter text, choose a style and add a badge or background. It is very much template-oriented but may be right if a simple text logo is all that is needed. Everything is done through the website, meaning there is no learning curve for users.
A slightly more advanced service is LogoMaker. This website allows the user to choose an icon from his own bank of images. The user then selects the colors he wants, adds text and downloads the finished logo. LogoMaker offers a website-ready logo for free, along with options to buy image files that can be used to make goods, such as cards and stationery.
In order to design a logo from scratch, a program such as GIMP is a possibility. GIMP is free to download and runs on Windows, OSX and UNIX. Although it is free, GIMP is a powerful option for a user who wants to design a logo. Its painting and image manipulation tools allow users to create professional-quality images. The GIMP website features a number of tutorials for new users, which can be very useful as the software has many features.