Let's put together a list of games like HOMAM - ie. turn-based strategy games with RPG aspects, heroes with skills and spells that command armies, can participate in quests and have a storyline in the campaign and some missions.

Age of Wonders (Can anyone tell me how does the game compare?)

Disciples I and II

Pocket Kingdoms and other games by that dev (they're basically HOMAM copies and even reuse some of their assets)

King's Bounty (it has a slightly different way of acquiring the armies though, depending on your rep and it might be way harder than usual games)

Feel free to suggest games that seem similar but are not necessarily turn-based.

Why HOMAM?
I'm pretty sure the known acronym that's been used for decades is HOMM.

Not necessary TB....Okay...Spellforce - best mix between RTS and RPG, inspired by the old Warlords Battlecry games, but vastly improving the formula.

Heroes of Annihilated Empires is another RTS/RPG mix, but isn't that great. For TB you also have the Warlords series and the original King's Bounty (which is the game that started the whole Heroes madness),

Disciples, Heroes and Age of Wonders have few elements that are common to all of them, but they do play quite differently. Age of Wonders relies heavily on how your units are placed and how they can use the terrain (especially in the last two games), Disciples requires to carefully select your hero build, what spells/items/troops you bring for a certain battle, while HOMM is more of a resource/numbers game.

Age of Wonders isn't really the same beast as HOMM at all. It has bits of Civ games, research, city building, an isometric/oblique battle system with troops that gain experience, etc. The older games could probably be more favorably compared due to their colorful palettes and surface basic mechanics, but 3 is essentially Civ 5 with magic, a better battle system and artifact weapons/armors/etc.

Also, you can summon armies of dire penguins. Yes. Penguins.

Ah, i totally forgot about penguins. Penguins are serious business, mate. Just look at Disgaea series:)

Here are two recent similar threads searching for turn-based strategy games:

Link 1
Link 2

Legends of Eisenwald would fit the bill.

Eador!
https://www.gog.com/game/eador_genesis
https://www.gog.com/game/eador_masters_of_the_broken_world
(Masters of the broken world is basicaly original with new graphics and few things changed)

Don't buy the remake, though. It's terrible.

The game I played the most was Master of Magic. I have little experience with HOMM, but it fits the description.
It is quite nice, but I can't compare them. Master of Magic seems more Magic and less armies, perhaps? It is great fun.

howso? why do you say that?

I was gifted the remake and don't have the original.

Last time I was playing it was pretty okay, a bit buggy, but playable. Since then there were many patches and some things added.
It's also important that you cannot normally save in the original (expect cheating with save directory).

I'll second this! I made random purchases of both and ended up loving them.

Thanks for all the tips! :)
Actually, because I wasn't aware/forgot there was a third installment of the games. Hehe! Because when I was into it and found out about the games, that acronym was being used. At least in the magazines and sites I read and visited.