In a recent article on Social Media Examiner, Ishita Ganguly discusses four ways to use social media to learn about your competition.
1 - Find shared fan interest - "Facebook Graph Search is a great tool for finding out what’s working for other brands on Facebook."
2 - Compare Facebook activity - "Fanpage Karma is a must-have tool for monitoring the Facebook activities of other brands in your industry."
3 - Analyze Twitter audiences - "Twitonomy shows you what’s working for other brands in your industry on Twitter. The tool also lets you track the major interests of other’s followers."
4 - Identify Google+ Influencers - "Google Ripples is an integrated Google+ feature for Firefox and Chrome users. You can use this tool to find power influencers on Google+."
We agree with Ishita's tips; they are excellent.
In addition, we suggest looking at what your competitor's campaigns on Instagram and Pinterest.
Lastly, there is an analytics tool called Simply Measured, that produces some amazing free social analytics reports.