On-the-run Treasuries are the most recently auctioned securities at each standard maturity (2-year, 5-year, 10-year, etc.). They are the benchmark issues that define the yield curve and trade with the tightest bid-ask spreads and deepest liquidity.
Once a new issue is auctioned at the same maturity, the previous on-the-run becomes off-the-run. Off-the-run securities are identical in credit quality but trade at a small yield premium (the "liquidity premium") because they are less actively traded.
The on-the-run / off-the-run spread reflects pure liquidity demand:
On-the-run Treasuries are important because:
The transition from on-the-run to off-the-run creates a predictable price dislocation that some relative value traders exploit.