Hi Alireza,

I'm not sure to understand correctly your question.

These extensions are sequence alignment formats. You can use samtools program (http://www.htslib.org/) for interacting with your data downloaded from ENCODE database.

hope that it help you

In general, the .bam file is an alignment of some experiment reads to a reference. BAM is the binary format, though they can be converted to SAM file if you want to read them. see samtools as suggested by Gael. 

The bam files you see on ENCODE are often FASTQ files (experiment reads) aligned to some reference (mouse, in your case), though this could depend on the experiment you're looking at. A .bai is the index for your bam file, it should have a matching name to your bam. It's used by tools to read your bam file. I like STAR as my aligner, but there are many different tools out there that will do a similar job (https://github.com/alexdobin/STAR/blob/master/doc/STARmanual.pdf), 

A "peak" is a region of interest from your reference that has been "called" by some tool. In the case of ChIP-seq experiments for example, we're interested in binding sequences of a protein. The peak is the approximate range the tool thinks this occurred. The narrow peak is a shorter peak than the broad peak calls, so you might want to use one or the other depending on your problem, or try both and compare results. I believe they're in a .bed file format, which is  chromosome, start position, stop position, and more fields (see https://genome.ucsc.edu/FAQ/FAQformat.html) You can use the peak for it's coordinate range, or to extract the nucleotide sequences contained in these peaks using additional tools (bedtools, for example) depending on what you're trying to do.

In addition to the above answers, which give you information about the different file formats, I think that you may be interested in ".narrowPeak" or ".bed" files. Those usually give genomic coordinates ("chromosome", "start" and "end" as a minimum) of regions which have been determined to show specific characteristics (enriched for TF binding as determined by ChIP-seq, open chormatin identified by DNaseI, MNase or ATAC-seq...). If you search through the ENCODE data, then you are probably "interested" in those peak or bed files. If you have a lot of loci to check and you know their coordinates, you can use the bedtools suite (http://bedtools.readthedocs.org) to check the coverage of your dataset by any other dataset in bed (or even bam) format... Good luck!