Multiple overlapping indices exist covering various investment universes. Almost all of the widely followed indices were originally created by Lehman Brothers and are now maintained by Barclays. The broadest U.S. dollar based bond index is known as the Universal. The Aggregate (often abbreviated Agg), which is historically the most popular index, more or less includes all bonds in the Universal rated investment grade. The direct analog to the S&P 500 would be the U.S. Corporate Investment Grade index, which is tracked by the ETF LQD, and contains exactly what it sounds like.
Citigroup (formerly Salomon Brothers) also has a competitor index to the Aggregate known as Broad Investment Grade (BIG), and Merrill Lynch (now Bank of America) has the Domestic Master.
Multiple other indices also exist covering other bond markets, such as international (non-USD) bonds, tax-exempts (municipal bonds), securitized products, floating rate, etc.