Characteristics: product length, GC content, base repeats, melting temperature, exon-exon junction vs including an intron, self and 3' complementarity, the specificity of the primers, primer length, GC clamp, etc....)
It really depends on what you're doing, but generally I use primer3 and check then check for secondary structure with the IDT oligoanalyzer tool.
Use the Oligoanalyzer to make sure the Tm of any secondary structure and self-dimer is much lower than your annealing temp (I don't believe Primer 3 checks for secondary structure, but could be wrong.)
The Tm should be around 60C- but it's more important that the Tms of both of your primers are within around 5C.
When considering product length, just make sure that you set your extension time (72C in the cycling parameters) long enough that it can make the entire product.