In Standard German, a phenomenon called terminal devoicing (Auslautverhärtung in German) affects the pronunciation of word-final (or more generally: morpheme-final) consonants. It leads to the merging of the phoneme pairs b–p, d–t, w–f, g–k and /z/–/s/ (a phoneme pair not reflected in orthography). These are typically pronounced as if the unvoiced letter were present. Thus, Rad is — in Standard German — pronounced as Rat (/rat/).
If the final consonant is moved away from the end of the morpheme, for example by adding a syllabic inflection, terminal devoicing does not kick in and the voiced sound is spoken. Thus, Räder is /ˈrɛ:dɐ/ while Räte is /ˈrɛ:tə/.
The morpheme-boundary one is important. If you build a compound word, such as Radlager, terminal devoicing still happens. Standard German pronunciation is /ˈratˌlagɐ/.
I explicitly said Standard German throughout this answer so far (in all cases except for one). This is because terminal devoicing is not prevalent throughout all dialects of Germany, Switzerland, Austria and the other places where German is natively spoken. Rather, it is a feature most present in the North of Germany. Southern speakers tend to differentiate between hard and soft final consonants, so that e.g. in Bavaria a difference can be heard between Rad and Rat.
Terminal devoicing is not a feature only present in German. It is also observed in Catalan, Dutch, Russian, Czech, Korean and others. The Scandinavian languages do not feature it, as it developed after the division between continental and North Sea Germanic.
: Some argue that the /a/ in Rad/Rat should be long (/ra:t/). I argue that long and short /a/ are more of a phonetic difference than a phonemic one in German.
: Beware with e.g. genitive forms. Des Rades is /ˈradəs/ because an additional syllable is introduced. The more common and shorter des Rads is /ˈrats/, because the consonant is still part of the morpheme-final consonant cluster.
: Southern here is meant as a synonym for oberdeutsche Dialekte: Bavarian (bairisch) and Alemannian dialects including those spoken in Switzerland, Austria, Alsace, South Tyrol and Liechtenstein. The Franconian dialects (which I consider to be central German, because Franconia is in central Germany) tend to merge the two sounds but much closer to a /d/ sound than a /t/ sound. You may realise that the entire issue is very complicated.
: Phonetically, one may argue that a final d may still be a tad harder than initial or medial d but the difference is less than the difference between d and t.