So, in Latin, there are 5 main cases* (and a couple more rare ones). They are nominative, genitive, dative, accusative, and ablative. Nominative is used mostly for subject and predicate nominative. Genitive is used to show possession. Dative, usually translated with "to/for", is used for indirect object and some other things. Accusative is used mainly for direct object and object of a preposition. Ablative is kind of a catch all case that can do a lot of things.
For more information, look here: http://dcc.dickinson.edu/grammar/latin/case-endings-five-declensions
In this sentence, the healthy professor is in the accusative, so he must be the direct object. Vinum can be nominative or accusative, but because professor is already the direct object, Vinum is probably the subject.
A declension is just a conjugation pattern. "First declension" nouns are the ones that end in "-a" ("femina," "sella") and conjugate based on that first chart. "Second declension" nouns (usually) end in "-us" or "-um" or "-er" ("vinum," "lupus," "magister") and follow that second chart. And so on.
Some second declension nouns are neuter and end in "-um" instead of "-us" in the nominative. Other examples are "templum" and "donum," which don't have that alternate form.
(These words will also end in "-a" in the nominative and accusative plural ("templa"), but otherwise the conjugation is the same.)