Odin is the chief god of the Aesir tribe. He is the god of many things and his influence is far-reaching. However, calling any of the heathen gods 'the god of this or that' is a vast oversimplification. As he is a god of war, as is Freya. The god of wisdom, but he takes advice from Mimir. The god of poetry, like Bragi, et cetra. This is something to keep in mind. In many of the things he has influence over, he isn't always the one you should turn to. He is certainly not the only one you could turn to.
He has many names which refer to his many aspects and should be used when petitioning him for a specific thing. In the eddas, sagas, and skaldic poems Odin has been referred to by literally a couple hundred names.
***Disclaimers: Many of these names are accented and pronounced much differently from how they appear in our alphabet. Look it up, and also dont use this as a sole reference. It isn't even close to a 'how to' guide. Just some information I noticed this place was lacking. These names are not strictly Norse. Some are derived from the Anglo-Saxon Woden and Germanic Wotann. the methods I mention for offerings are not the only ways to do it. I realize this reads like the side effects list on a medicine bottle and I apologize. Everything but the names and their translation are purely my opinion and experience. These are not even close to all the names. Just the ones I know off hand at the time of writing this. Finally, I can't stress this enough - RESEARCH. Odin does not suffer lazy fools.
He is Ofnir - 'The Inciter' or 'The Inspirer.' both refer to him being the patron of berserkers. Ofnir should be appealed to through symbols and acts of violence such as fighting, weaponry, blood, physical ordeal, and emotional ordeal that results in rage. Practicing the berserkergangr is also helpful. In my experience, calling on Ofnir is extremely effective in helping to bring on the gangr.
He has several names pertaining to being a god of warfare in a more general sense than Ofnir. such as Valdadr - 'Father of The Slain, Sigfadr - 'Father of Victory,' Biflindi - 'The Spear Shaker,' Bodogaedir - 'Battle Enhancer (or something similar),' Dorrudr - 'Spearman,' etc, etc. Calling upon Valfadr et al. should be approached in similar fashion to Ofnir, but with less emphasis on rage. He can be called upon for aid in battle.
He is Gangleri - 'The Wanderer.' who can be called on for aid in a journey, moving, travel in ways such as preparation, not getting mugged (personal experience speaking there - should this happen anyway, see 'Ofnir.'), etc. Appealing to him can involve travelling - anything from walking down a road in town you've never been down before to buying a one way plane ticket to the other side of the world and 'seeing what happens.' and the Raido rune.
He is a god of magic in several ways. Runatyr - 'Rune God,' Gondlir - 'Wand Bearer,' and Galdrafodr - 'Father of Galdr,' are all fairly self explanatory. For works in concealing your magic workings (or deception in general), he can be called on as Grimnir - 'Masked' or Fjolnir - 'Concealer.' I personally have never tried this. Some names can be called on to aid with 'offensive magic,' also. Such as Bolverkr - 'Evil Worker/Doer.'
For wisdom, he can be called on as Svidurr or Fjolnir - both meaning 'Wise One,' and Harri - Wise One. These are not a substitute for research and experience. But they can be used to help facilitate a deeper understanding of the runes, magic in general, people (arguably), etc.
Some names simply refer to his appearance, deeds and relation to other gods. like Harbard - 'Greybeard,' Hangatyr - 'Hanged God,' Hoarr - 'One-Eye,' or Borsson - 'Son of Bor.' I typically appeal to any aspect of Odin in verse. Since he's also a god of poetry (Sadr - 'Sooth') and I tend to use these as a way to signify that I'm not JUST applying to, say, Ofnir; and also because using the same name doesn't line up with Skaldic verse (for example, in Ragnar Lodbrok's saga, a giant snake is referred to as several things such as 'Heather-Eel,' 'Ground-Wolf' and 'Earth-Coil). Plus it sounds better than saying "Ofnir" a dozen times.
An interesting one I've come across whose old norse equivalent I've forgotten is, in English, 'Horse Hair Mustache.' just a fun fact.
Finally, there are multiple variations on each name that can be used. For example, Hangi=Hangagud=Hangatyr. (Hangi means 'Hanged One' and Hangagud and Hangatyr both mean 'Hanged God').
This should go without saying. But don't be an idiot and attack random people, freeze to death in the woods, or get yourself stranded on the other side of the earth. seriously.
Names of Odin