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Fighters are heroes who go about hitting evil things for a living. They're generally good at surviving long enough to brag about it afterwards.
Because of the fighter's Veteran trait, he can very easily achieve multiple level-ups in short succession with his popcorn. This synergizes greatly with the Elf's expanded mana pool, allowing the Elf Fighter to rival even the Gnome Sorcerer in mana-spiking capability.
While the Goblin Fighter lacks the mana of the Elf, it makes up for it with even more frequent level-ups. With WONAFYT and BLUDTUPOWA, the goblin fighter can practically ignore the map past the early levels, and focus solely on dumping every resource on the map into killing the boss.
The fighter levels up significantly faster than other classes, which has two huge benefits. Being a higher level he is more powerful and dangerous, but it also benefits more often from the full-heal and mana-restore you gain when you level up. This extra leeway in terms of experience gain and leveling up makes the Fighter a competitive class. His other class features are largely inconsequential. Pit Dog is a one-time event, and basically equivalent to one use of the CYDSTEPP glyph. Instinct is completely useless, since fighters want to avoid fighting monsters that are equal or lower level than themselves if at all possible, saving them for level-up catapults later in the game.
Although he's presented as a Fighter, this character is actually equally well-suited to being a spellcaster. He has no class features that help his melee combat skills, and largely comes across as a tabula rasa that can be played any way you want. It's always a good idea to keep an open mind with the fighter and be as flexible as possible based on what items and glyphs you find.
This is the first class you are presented with when you begin Desktop Dungeons, and he functions very well as a beginner class, and also presents a strong foil to the specialized classes such as Warlord and Assassin that are unlocked later.
As for deities, Tikki Tooki is an incredible choice. The fighter will be able to rack up a ton of piety, as he will reach a high level quickly and leave a bunch of popcorn untouched. The massive piety gain allows him to invest heavily into into Reflexes and Tikki's Edge, which combine to massively increase the fighter's already formidable level-up spiking potential. Glowing Guardian also works well: the Fighter's XP bonuses synergise very well with the Humility boon, and the Fighter also has enough spare XP to make liberal use of Absolution. The Fighter works well enough with any other deity as well, so for the most part you should adapt your worship strategy to the needs of the dungeon.