Datasets of aligned nucleotide sequences
You can find 46-way multiz alignment from UCSC genome browser, it is down on comparative genomics part and labelled as "cons 46-way", which is a genome alignment of 46 vertebrate species. You can use data on their genome browser on the site, or get download information here.

If you are interested in pair-wise alignments, I don't know of any pair-wise alignment database, but in fact you don't need one. You can search for nucleotide sequences from NCBI nucleotide database and align them using BLAST on their website. BLAST is maybe the most common tool for pair-wise alignments and also for database alignment searches, where a single query sequence is searched for matches throughout a database of sequences. If you want to do a large number of alignments, you can download BLAST to your computer to do them faster.