How is a portion of DNA selected and unwound from nucleosome?
This answer will be a very broad overview and is based largely on information from the textbook "Molecular Biology of the Gene" by Watson et al. (which I highly recommend). 

Nucleosomes are dynamic structures.

The interactions between DNA and histones are non-specific and dynamic. Regions of DNA can be transiently unwrapped from the nucleosome and thus recognized by DNA binding proteins. In fact, because of the non-specific nature of the interaction, DNA can slide relative to the histone core and expose different sections of DNA. This can be facilitated by nucleosome remodelling complexes which can also catalyze complete ejection of histone octamers to expose even more DNA. 

Nucleosomes can be positioned specifically.

The association of DNA binding proteins with specific sequences can prevent or control nucleosome formation at specific loci and leave tracts of exposed DNA accessible to the transcription machinery. Additionally, certain DNA sequences that form bent helices are preferentially bound by nucleosomes. 

Chromatin compaction is highly regulated.

Two broad classes of chromatin are present in the interphase nucleus: highly condensed, transcriptionally inactive heterochromatin and less condensed, transcriptionally active euchromatin. Chromatin compaction is regulated by post-translational modification of histone tails, and modifications such as acetylation and phosphorylation, especially of the N-terminal tails, can prevent higher order, repressive chromatin compaction. The specific pattern of histone modification is referred to as the histone code and certain modifications are associated with gene expression. These modifications can be recognized by remodelling complexes and even transcription factors. 

Further Reading:

Becker PB, Workman JL. 2013. Nucleosome Remodeling and Epigenetics. Cold Spring Harb Perspect Biol 5(9):a017905.