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Fahrrad vs. Rad

Word practice demo
In German, both Fahrrad and Rad translate to bicycle. Fahrrad is the formal and complete term, whereas Rad is a very common, shorter, and more colloquial alternative, much like the difference between bicycle and bike in English.

Fahrrad

A1
This is the standard, official, and complete word for bicycle. It is a compound noun made from fahren (to ride/drive) and Rad (wheel). You will find this term in written texts, on official signs, and in situations where clarity and formality are desired.
Das Abstellen von Fahrrädern ist hier verboten.
(Parking bicycles is forbidden here.)
Ich habe mir zum Geburtstag ein neues Fahrrad gewünscht.
(I wished for a new bicycle for my birthday.)
Ein guter Fahrradhelm ist beim Fahren sehr wichtig.
(A good bicycle helmet is very important when riding.)
Der Diebstahl von Fahrrädern hat in der Stadt zugenommen.
(The theft of bicycles has increased in the city.)
In diesem Geschäft kann man Fahrräder mieten.
(You can rent bicycles in this shop.)

Rad

A2
This is the common, everyday shorthand for Fahrrad. It is used very frequently in spoken German and informal writing. While Rad also means wheel, the context almost always makes it clear that a bicycle is meant.
Ich fahre jeden Tag mit dem Rad zur Arbeit.
(I ride my bike to work every day.)
Wo hast du dein Rad angeschlossen?
(Where did you lock up your bike?)
Mein Rad hat leider einen platten Reifen.
(Unfortunately, my bike has a flat tire.)
Lass uns die Räder nehmen und zum See fahren.
(Let's take the bikes and ride to the lake.)
Das Vorderrad an meinem Rad eiert ein bisschen.
(The front wheel on my bike wobbles a bit.)

Summary

In summary, Fahrrad is the complete and more formal term for bicycle. Rad is the widely used, shorter, and more casual equivalent. In most everyday situations, they are perfectly interchangeable, with Rad being more common in spoken language. The choice is similar to using bicycle (more formal) versus bike (more casual) in English. Although Rad can also mean wheel, the context almost always prevents any confusion.