What does a red icon mean in TF2 on the killboards?
Solution 1:
If the background is red, it means that the weapon killed the player with a crit.
So in your first screenshot, it could be read as:
soccer sam88Phlogistinator (crit)killed Table
whereas the second screenshot would read as:
didn't notice youUbersaw (noncrit)Neil Armstrong
As an additional note, as mentioned by users in the comments, the background for mini-crits looks exactly the same as if you had killed the player with the critical hit.