what does it mean

[Fiona Forris]

end of my tether

[Moog]

A tether is a piece of rope attached to a fixed point usually used to restrain animals or prevent hot air baloons taking off, when you reach the end of your tether [i.e. the end of the rope] you can go no further.

[andy hughes]

It translates as someone who has had enough of a particular circumstance, and is about to loose his or her temper. It comes fromthe days when tethered animals, if excited or upset, would strain to escape their tether, and thus, stretch it as far as it would go, and be 'at the end of their teather'.