Does it affect my standing with the faction, Federation rank, that kind of thing?
It depends on the mission but generally, yes!
When you accepted a mission you may have noted that it already listed a penalty for failing it. Should you fail, this penalty will be issued as a fine (which may become a bounty if you let it expire).
In addition, missions are associated with one minor and possibly a major faction. Minor factions include, for example, the Sirius Corporation, while major factions include, for example, the Federation. Should you fail a mission, the associated factions may loose some of their influence, depending on the value of the mission. In addition you will loose some of your standing with them (reputation). Note that your rank, once attained, remains unaffected as you cannot be de-ranked.
In addition, some missions, notably those associated with Powerplay, may impact the preparation, expansion, and maintenance phases of a faction should you fail a mission associated with these. This in turn may have an impact of your ability to gain merit with the power or may, eventually, even cause your faction to drop out of Powerplay altogether should it fail its cycle goals.