What is the rule for using an -er ending or a "more" beginning.
Ex: She is happier than I. v. She is more happy than I.

Please let me know.

The general rule is that er is used when the word is one or two syllables. More is used when the word is three or more syllables.

Happy - Happier, Happiest
Efficient - More Efficient, Most Efficient

From bartleby.com --

There are two patterns for comparison of adjectives, the inflected and the periphrastic. The inflected pattern adds -er to the positive degree of the adjective to form the comparative degree: small becomes smaller, happy becomes happier. To form the superlative degree, it adds -est: smallest, happiest.

The periphrastic pattern uses the adverbial intensifiers more and most: the comparatives of beautiful and ostentatious are more beautiful and more ostentatious; the superlatives are most beautiful and most ostentatious.

The generalizations that seem to account for whether we choose the inflected pattern or the periphrastic are these:
(1) most one- and two-syllable adjectives use the inflected pattern;
(2) adjectives of three and more syllables almost always use the periphrastic;
(3) the higher the frequency of two-syllable adjectives, the more likely they are to inflect for comparison;
(4) the periphrastic more and most may on occasion be used with any one-syllable or high-frequency two-syllable adjective, e.g. more dear, most happy.