"I know your horse drinks wine."
Translation:Jeg ved din hest drikker vin.
That is very close. I'm going to take the liberty of translating from this website. For the verb "kende" (know) we have the noun "kendskab" (familiarity). For the verb "ved" we have the noun "viden" (knowledge). Normally "viden" is perceived as a deeper insight than "kendskab", which can perhaps be more superficial. In practice "ved" often takes an object that is a complete sentence, while "kende" often takes an object, which is a pronoun or a noun.
"Jeg kender hans adresse" (I know his address) /// "Jeg ved hvor han bor" (I know where he lives)
"Jeg kender ham godt" (I know who he is) /// "Jeg ved hvordan han er" (I know how he is - in this case we're not talking about well-being, but what kind of person he is)
"Kender du dansk kultur?" (Do you know Danish culture?) /// "Ved du noget om dansk kultur?" (Do you know anything about Danish culture?)
Hope that helps!
Good reply on the whole, "kende" refers to familiar knowing, cognate with "kennen" in German. "vide" is cognate with "wissen" in German and refers to intellectual knowing. The actual use is slightly nuanced as jackinaboxx described with respect to the German equivalents. I should point out that both are transitive verbs, "vide" doesn't need to take a clause, it can take a singular noun. You'll most often hear Danes using it in an admission of ignorance, for instance "Det ved jeg ikke"!