How is the alpha carbon defined in alkyl halides? The &alpha; carbon refers to the first carbon that attaches to a functional group. According to this information, the alpha carbon in 2-bromobutane should have been the carbon next to the one with bromine. But in my book it says that the alpha carbon in that compound is the one with bromine atom. How is this possible?
The &#x3b1; carbon is the one directly attached to the functional group. There is no contradiction to your book’s claim that the &#x3b1; carbon of 2-bromobutane is the one directly bound to bromine.

This is due to the different definitions of functional groups and what they include. The keto functional group is defined as $\ce{-C(=O)-{}}$, i.e. it includes the carbon atom doubly bonded to oxygen. Halogen functional groups, for example the bromo functional group are defined as only the heteroatom, so in this case $\ce{-Br}$. Note that because the carbon is not included in the definition of this functional group, it is instead the &#x3b1; carbon.

This means that reducing the ester functionality of a &#x3b2;-halo ester will result in a &#x3b3;-halo alcohol even though no group migrated.