Those would be literal translations and they're not wrong at all, but (at least in my area) I never hear anyone use "to please" or "pleasing." I think translating "jdm gefallen" to "like" is a more natural translation and that's how I learned the use of "gefallen." I think it all depends on how you want to translate: directly or naturally.
This is such a great question and the link I will provide at the end of this comment provided me with a great explanation that really helped explain the difference. It's a bit lengthy but it's such an important part of the German language that it's worth the read.
To summarize (and this may have been answered in the comments below but it never hurts to have multiple explanations):
Mögen (verb): this is typically used for general declarations of "liking" something -- specifically a noun or pronoun. ,,Ich mag die Kekse!" ,,Wir mögen deine neue Freundin." Doesn't work well for "liking" activities (aka other verbs). ,,Magst du zu essen?" is an example of this verb used incorrectly (and that's probably just bad grammar anyway). ,,Mögt ihr meinem Backen?" "Nein, wir mögen deine Kekse gar nicht." Sad face.
Gern/e (adverb): If you want to express that you enjoy doing a particular activity (AKA: another VERB), such as sleeping, this is your word. ,,Ich schlafe gern in meinem eigenen Bett!" Since it's an adverb, you would position as you would an adverb -- usually directly after the verb in the second position, or before an infinitive. ,,Isst du gerne mit Essstäbchen?" (someone please verify the validity of my examples..). Gern vs Gerne: it doesn't matter if you use the trailing E or not, it has the same exact meaning and just serves to modify the sound of the word and in effect, rhythm of the sentence.
Gefallen (verb): This one is used to express that something is "pleasing to you." Instead of saying "i LIKE something"...we are reversing the roles and saying that "something IS PLEASING TO me." This is why we usually see it conjugated in the third person form: gefällt. zB: ,,Der Sonnenuntergang geffält mir." It takes a dative case so that's why we see the "mir" -- since the object that's pleasing is acting its verb out on the speaker. It's more of a statement about the object than about how you feel, emphasizing some intrinsic quality in the pleasing thing and not your own feelings. Apparently this is only used for things tied to sight/hearing, and not taste/touch/smell (for whatever reason). The link explains it detail and has more examples.
There's more to it and the this goes into much more depth on all three:
This blog is usually a good resource for these types of difficult questions because they are explained so thoroughly.
This is a great explanation. If you don't know what "gefallen" means to begin with, the hints are not very helpful. I think if they included "is/are pleasing", it might help some of us figure out the meaning better. I keep messing this up (but that's mostly because somehow I keep thinking that "gefallen" means "to happen" - maybe I'm thinking of the English "befall" and that's nobody's fault but my own.) I guess I'll have to write this one out ten times in longhand to get my brain to accept it.
I didn't think of it like that, you're right. It would probably be good for beginners to think of "gefallen" as "to please sb" at least because it would help them remember to use the dative noun/pronoun.
I guess I just never thought there was much of a difference between "mögen" (except for it's use in the konjunktiv - "möchte," etc.) and "gefallen." I've used them pretty much interchangeably when writing to my penpal and haven't been corrected for using one rather than the other. To me, "gefallen" generally sounds a little better than "mögen," but I can't quite think of why (maybe "mögen" sounds simpler to me? That might just be because it's one of the first words I learned/used in German). "Gern" is easy to tell apart though because it's used for when you like doing something as opposed to liking a thing, z.B. "Ich spiele gern Fußball" - "I like to play soccer."
I know what you mean but the problem is that it's hard to unlearn things once you've learned them incorrectly so if Duo were to do this with everything, people learning English would think that we talk like this! We don't. A lot of people learn languages 'from English' because the new language isn't available to learn from their own language so I think the correct English should be given as opposed to the literal meaning.
The easy books please him. Ihm gefallen die einfachen Bücher. The conjugation of "gefallen" needs to agree with the subject, which is the books. He (him) is receiving the action. That's one thing to remember with gefallen; it's the thing that is doing the pleasing that is the subject and the verb is conjugated to agree with it. (The other thing to remember is that gefallen requires the dative for the object, thus ihm instead of ihn, which you have correct.) You can rearrange the words somewhat, but it doesn't change the needed conjugation.
To keep things clear when forming these sentences it might help to keep the subject first:
Der Hund gefällt mir. Die Hunde gefallen mir.
Die Blume gefällt ihr. Die Blumen gefallen ihr.
Das Haus gefällt uns. Die Häuser gefallen uns.