Backlinks are simultaneously one of the most powerful and most problematic tools when trying to optimize a domain via SEO. First, a bit of introductory to what, exactly, a backlink is. Basically, a backlink is just an inbound link to a web page; for example, whenever someone accesses a website, that's a backlink. The strength of backlinks is also their weakness; search engines tend to put a heavy emphasis on the number of backlinks a webpage has when listing it in their search results. A page with a higher number of backlinks will be weighted heavily above a similar page with fewer backlinks. Google's PageRank system is an example of a system that uses backlinks to try and analyze the worth of a page via backlinks. The problem here is that backlinks are just used for everyday SEO purposes- several spam oriented companies use this knowledge to rig the system and create problems.
The process is referred to as linkspam; the idea is to really enhance one page's number of backlinks by creating a number of other pages that link to that page, regardless of content or whether or not the pages are even related in any significant way. This, however, is a dying trend. Once upon a time backlinks basically counted as a single vote for a webpage. Recently, however, as a way to combat linkspam various methods of balancing the value of a backlink have been implemented. Basically, the idea has become to check the relevance of a backlink to the content of the page it has been linked to; if the pages don't have any tangible relation, the value is reduced significantly. Additionally, too many backlinks over a short period of time (a day, for example) is considered suspicious and can result in a site being penalized or removed from the search results entirely.
All of this information is relevant to you as you look to purchase an expired domain. There are services and tools out there that exist solely to research the history of a domain name, and of course an important piece of that history is the number of backlinks. If an expired domain has a healthy number of backlinks, then it will have an edge up over the competition. If, however, it seems to have been connected to linkspam, you will want to avoid that domain as it is very likely penalized and will be counterproductive to your needs. The key is to utilize one of these tools to get a summary of backlinks and try to find a successful domain.