A degenerate primer is mixture of primers that has substitution of different bases sequence (they are similar not same).  They are usefull if need to amplify a gene from similar organism. So it possible amplify different  sequence which represent different protein sequence. They maybe used when need to design primer based on protein sequence. As we know different codon codify for different protein.

Universal primer is a sequence (single) used for the amplification of a similar gene that related to a specific  Genus. In bacteria we need it to amplify ribossomal RNA. For example to amplify enterobacteriacea gene we need universal primer for it. That way all species of enterobacteria are amplified (eg: Escherichia, Salmonella)

http://online.liebertpub.com/doi/pdfplus/10.1089/cmb.2005.12.431

http://bitesizebio.com/18992/a-primer-for-designing-degenerate-primers/

A degenerate primer is mixture of primers that has substitution of different bases sequence (they are similar not same).  They are usefull if need to amplify a gene from similar organism. So it possible amplify different  sequence which represent different protein sequence. They maybe used when need to design primer based on protein sequence. As we know different codon codify for different protein.

Universal primer is a sequence (single) used for the amplification of a similar gene that related to a specific  Genus. In bacteria we need it to amplify ribossomal RNA. For example to amplify enterobacteriacea gene we need universal primer for it. That way all species of enterobacteria are amplified (eg: Escherichia, Salmonella)

http://online.liebertpub.com/doi/pdfplus/10.1089/cmb.2005.12.431

http://bitesizebio.com/18992/a-primer-for-designing-degenerate-primers/

In addition to António's answer, PCR/sequencing primers that bind to a sequence found in many plasmid cloning vectors, most of which are derived from pUC vectors, are referred to as universal primer. These sequences are defined as good PCR and sequencing sites as they flank the multiple cloning site. Universal primers are really not 'universal' in the sense that they will bind to anything.

Thank you! 

vDegenerate primers

•It’s a combination of oligonucleotide sequences in which few bases are altered in such a way that the primer covers the all possible nucleotide combinations for the targeted protein through the DNA sequence

•ATCGTT[GC]AAGT[AGC]ATC : GC= S, AGC=N

-Universal primers

•Anneal universally to organisms and amplify.

•conserved region is used

•Eg.  Universal primers are used to amplify the 16SrRNA gene from the rumen metagenomic DNA (Weisburg et al., 1991), RNA isolation