Sequencing a specific region of a genome
Yes, it is possible to sequence a specific region of the genome. The method, as you mentioned, is called targeted sequencing. Resequencing is basically sequencing something that has already been sequenced. This means that instead of assembling all of your sequence reads from scratch, you can just align them to the reference sequence (in your case, the entire human genome has been sequenced). Targeted resequencing means you're sequencing a specific region, such as a gene. This requires concentration of the specific DNA you wish to sequence and can be done by PCR amplification (using primers that flank your desired sequence) or hybridisation (using probes complementary to your desired sequence that are fixed to a surface), among other methods. 

For information on methods for enriching a DNA sequence, give this a read:

Mamanova, L. et al. Target-enrichment strategies for next-generation sequencing. Nat. Methods. 7, 111-118 (2010).