Taegeuk (sometimes spelt taeguk) patterns form the backbone of the grading system for WTF (World Taekwondo Federation) Taekwondo in the same way Kata is practised in Karate. There are eight WTF patterns (Taegeuks) for Kup grade (coloured belt) students to learn and demonstrate during gradings so that the student can display their proficiency in individual techniques and the ability to perform techniques in a logical sequence. There are further patterns for Dan grade (black belt) beginning with Koryo. The Taegeuk patterns are all intended to simulate multiple attacks coming at the student from all sides.
The Taegeuk patterns become progressively more complex, and introduce more advanced techniques the higher you go. Taegeuk Il Jang (the first pattern, said to symbolize "Heaven and Earth") is the simplest employing only the bare basic techniques. Taegeuk Pal Jang is the eighth and includes much more complex techniques, and more complex ways to string them together. Many people regard the 6th Taegeuk pattern (Yuk Jang) to be the most complex and difficult of the Taegeuks, and possibly more difficult than some Dan grade patterns.
The Taegeuk patterns were originally intended for use by junior grades studying the WTF Taekwondo system. Over time, they were adopted as the basis of the grading syllabus for adults as well replacing the Palgwe forms, which are still in use by traditional ITF Taekwondo schools. The Taegeuk patterns are described in David Mitchell's Official WTF Taekwondo (a great reference for any student) as "Basic Taegeuks for practice". I would say they (particularly the later patterns) are anything but simple. To get them right requires a great deal of practice and understanding of the various techniques.
It's important to note that once a pattern is performed at a successful grading, you carry on learning that pattern throughout your Taekwondo training. The examiner may well ask to see Taegeuk Il Jang (or any pattern previously learnt) at any one of your higher level gradings, including Dan gradings.
There are entire books available giving advice on how to perform patterns correctly, and of course an instructor will always have his/her advice for the student on how to improve a pattern. I would offer these simple words of advice for good execution of Taegeuks: