"Your father, on the other hand, favored a mahogany wand. Eleven inches. Pliable. A little more power and excellent for transfiguration. Well, I say your father favored it -- it's really the wand that chooses the wizard, of course."
-- Mr. Ollivander (PS5)
According to Mr. Ollivander, James Potter had a mahogany wand, eleven inches, pliable, and “excellent for transfiguration” (PS5). It makes sense that a wand good for transfiguration would be “pliable,” which means that it can be molded and shaped. And this wand was quite successful since James became an Animagus stag while at Hogwarts (PA18).
Harry’s first broom was a Nimbus 2000 with a mahogany handle given to him by Professor McGonagall for the Gryffindor Quidditch Team (PS10), It was destroyed by the Whomping Willow in his third year (PA9).
It's probably no coincidence that Harry's father James had a wand of mahogany, the same wood as his first Nimbus 2000 broomstick since part of Harry's legacy from his father was athletic ability. James played for Gryffindor as a Chaser (Sch2), and Lupin says Snape was jealous of his "talent on the Quidditch field" (PA18). In an old photograph James watched baby Harry fly around on a mini-broomstick Sirius Black bought him for his first birthday (DH10).
The fact that James Potter was "chosen" by such a rare and costly wand wood seems to be in keeping with his upbringing as the "very pampered ... extra-treasure" only child of pureblood parents (TLC). Upon their deaths, James also inherited "plenty of money" (AOL) which he passed along to Harry in a vault at Gringotts, not to mention an heirloom Cloak of Invisibility that was one of the Deathly Hallows passed down from Ignotus Peverell (DH35).
From the Web
Mahogany is a finely grained, reddish-brown wood of the tropics used to build expensive furniture, ships and musical instruments (Wikipedia).
Writing by J K Rowling on Pottermore:
Pottermore feature: Wands
Harry Potter Wiki: Mahogany