In World of Goo, you control Goo Balls which you can link together in various ways to form structures.
In most levels, you can simply drag and drop the balls to link them into a structure. You normally need to build in generally triangular structures as most Goo Balls require at last two arms to grab on to other balls.
Special types of Goo Ball exist that only have one arm (water for example), or that are lighter than air (like balloons). Others cannot be attached but can build towers. Still more balls can be shot in a direction but not dragged. There are dozens of types of ball in the game, but you are introduced to each gently.
Most levels require you to rescue Goo Balls to complete them. You must build towards the level's pipe so that it can suck the remaning balls. If you can't see the pipe, a small arrow will on the screen will be pointing toward it.
Some levels do not have pipes. In such levels you are given clues as to where you have to build or what you have to do.
The Sign Painter is your friend. Almost every level has signposts (which are animated so you can see them easily). Click on them and he'll give you a clue.
Obsessive Completion Distinction
For those in need of further challenge, every level has an "OCD" mode. Press ESCAPE to see the game menu, and click "ocd" to see what the additional requirements are to obtain OCD. They involve rescuing additional balls, or completing in a limited number of moves or amount of time. When you meet the OCD requirements of a level, a little flag will appear on the map screen besides that level.
You first encounter time bugs in Impale Sticky. They are little white flying bugs that you can can click on to "undo" one move. Be careful how you use them though, for you won't always get a new time bug when you make a move.
Once you have completed the minimum level requirements, all time bugs will vanish, which can be annoying if you're trying for OCD.
Once you've completed Leap Hole in Chapter 2, you gain the use of a whistle. Hold down the mouse button to whistle and nearby Goo Balls will be attracted to it. They'll move faster and tend to head towards the sound.
The whistle is useful to get balls where you want them quickly; for example to stabilise a structure by changing the weight distribution, or to get balls to the exit quickly.
Once you have the whistle you can use it on any level, including earlier levels if you choose to go back to them to attempt OCD.