The Difference Between Hoping and Hopping

The words hoping and hopping are both present participles (verb forms ending in -ing), but they're pronounced differently and their meanings are not related at all.

Definitions

Hoping is the present-participle form of hope—to wish or feel that something good or desirable will happen, or to expect something with a degree of confidence. 

Hopping is the present-participle form of hop—to make small jumps (sometimes on just one foot), to be very busy, or to be extremely upset (as in the expression "hopping mad").

Examples

• Jimmy left for school early the next morning, hoping to avoid Benny and his buddies.
• Kolya was hopping along like a chicken trying to catch a worm.
• "I had no appetite for the sparrows hopping from tree to tree above me, but there seemed no way to convince them of that. Each one, so great is his vanity, thinks himself eminently edible."
  (J.F. Powers, "Death of a Favorite." The New Yorker, 1951)
• "Bo shuffled along, quickly, almost hopping, hoping to beat her to the door."
  (Thomas Glynn, "Bo and Be." Statements 2: New Fiction. Fiction Collective, 1977) 

Usage Notes and Idiom Alert

"When a verb ends in a final silent e, drop that e before adding the suffix -ing. The word is hope with a long o sound. The final silent e makes it long. It is not hop, which is pronounced with a short o and is what bunnies are supposed to do...
"Hop is a one-one-one word: a word of one syllable, ending in one consonant, preceded by one vowel. You double the final consonant before adding a suffix beginning with a vowel."  (Elizabeth Hagner, Spelling Demons Week by Week. Walch Publishing, 1997)

Hoping Against Hope

"[To 'hope against hope' means] to hope or wish for with little reason or justification, as in 'I'm hoping against hope that someone will return my wallet.'"
(Christine Ammer, The American Heritage Dictionary of Idioms, 2nd ed. Houghton Mifflin Harcourt, 2013)

Practice

(a) Mary saw Paul _____ along the pier.
(b) She was _____ that he wouldn't trip.
(c) "The restaurant was _____—Grant had told her about its growing popularity—but fortunately they had reservations."  (Jennifer Lane, Bad Behavior. Omnific Publishing, 2011)

Answers to Practice Exercises

(a) Mary saw Paul hopping along the pier.
(b) She was hoping that he wouldn't trip.
(c) "The restaurant was hopping—Grant had told her about its growing popularity—but fortunately they had reservations."  (Jennifer Lane, Bad Behavior. Omnific Publishing, 2011)