I am wondering if there is a term for an adjective that modifies a preceding subject (e.g. I am happy.) and one that modifies what follows (e.g. I am a happy person.).
The comments by user2619 and jlawler answer the question directly. I can add more information here, so that a deeper understanding of the distinction between predicative and attributive adjectives is established.
A predicative adjective is (part of) the main clause predicate, hence the term predicative. An attributive adjective is also a predicate, but one that is not part of the main clause predicate, e.g.
a. The happy boy is hungry.
The main clause predicate is the two-word combination is hungry, so hungry is a predicative adjective. The attributive adjective happy is also a predicate, but it is not part of the main clause predicate, but rather it is a secondary predicate (or embedded predicate). Both adjectives are assigning a property to the boy, so both are definitely predicates in the relevant sense.
From a technical standpoint, the distinction between attributive and predicative adjectives can be examined from the point of view of structural syntax. An attributive adjective is a dependent of the noun to which it assigns a property, whereas a predicative adjective is not a dependent of the noun to which it assigns a property.
Many languages encode the distinction in terms of word order, as pointed out in the question. In Germanic languages (English, German, Dutch, Icelandic, Swedish, Norwegian, Danish, etc.) attributive adjectives precede the noun to which they assign a property, whereas predicative adjectives usually follow the noun to which they assign a property.
While predicative adjectives are often introduced by the auxiliary verb be, they also appear without be, e.g.
b. That made me angry. c. They arrived drunk. d. Tom sat in the corner upset.
The adjectives angry, drunk, and upset are predicative adjectives; they follow the nominal to which they assign a property.
Finally, some languages encode the distinction between attributive and predicative adjectives morphologically. In German, for instance, attributive adjectives bear suffixal endings that show gender, number, and case, whereas predicative adjectives lack these endings entirely, e.g.
e. ein betrunkener Mann - 'a drunk man' f. Der Mann ist betrunken. - 'The man is drunk.'
The -er ending identifies betrunkener in e as an attributive adjective, whereas the lack of an ending on betrunken in f identifies it as a predicative adjective.