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A gerund must end in -ing, Be a verb form and act as noun in a sentence.

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It's a verb that can be used as an adjective.

There are two type of participles and there are present participles and past participles.

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Infinitive take many different grammatical forms:

Nouns, adjectives, adverbs, but does not function as a verb.

Formed by taking the word “to” and adding it to the base form of a verb: to walk, to be

express an action or a state of being

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My dog greatly enjoys Barking at the mailman …and anyone else that walks by the house.

Is “Barking” gerund, then it must have:

Ends in -ing: yes

Verb form: yes

Acting as noun:yes

Then Gerund phrase of this sentence is : “Barking at the mail man.”

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Training is hard work, but worth it when you accomplish your goal.
Is "Training" gerund then it must have:

End in -ing:Yes

Verb from:Yes

Acting as noun:Yes

Then gerund of this sentence is : "Training"

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The smiling graduate threw his cap in the air.
[Smiling end in -ing, and it, formed from a verb smile, modifies the noun graduate.]

Present participles end in -ing.

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For dinner, we had grilled steak and mashed potatoes.
[Grilled, formed from the verb grill, modifies the noun steak;
mashed, formed from the verb mash, modifies the noun potatoes.]

Most Past participles end in -d or -ed.

Some are formed irregularly.

e.x:, break/broke/broken

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Infinitives function as adjectives when they modify or describe nouns in a sentence.(give us more information about nouns)

Example: Yuka wants a pizza to eat
“to eat” describes or gives us more information about the pizza.

And Yuka isn't looking for just any pizza, she's looking for a pizza to eat.

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Infinitives function as adverbs when they are used to give more Information about adjectives, verbs, adverbs in a sentence.

Example: She raised her hand to ask a question.

“To ask” giving us more information about the verb (raised).

Why did she raise her hand?
She raised her hand to ask a question.
Giving more information.