Although the Valar may seem inactive through the later (i.e post-First Age) history of Middle-earth, they never actually abdicated their roles (although, as we'll see, their capability for direct action does decrease).
"Valar" means "powers", and the Valar are the Powers of the World. Their role was initially to prepare Arda for the coming of the Children of Ilúvatar, then to manage Arda in a more general sense.
This management of Arda is an ongoing job, but one which requires less direct intervention over time (see below), and the Valar only ever laid down their government of Arda once, as is told in the Akallabeth:
Then Manwë upon the Mountain called upon Ilúvatar, and for that time the Valar laid down their government of Arda.
It's notable that the text here specifically states "for that time" - in other words, the Valar took up their government of Arda again after the destruction of Númenor.
Regarding Manwë, yes, he really does just sit and wait for the End; this is confirmed by a note in the Myths Transformed essays in History of Middle-earth 10:
Nonetheless the breaking of Thangorodrim and the extrusion of Melkor was the end of 'Morgoth' as such, and for that age (and many ages after). It was thus, also, in a sense the end of Manwë's prime function and task as Elder King, until the End. He had been the Adversary of the Enemy.
A further note in the same set of essays has this to say in a more general sense about the Valar:
The Valar 'fade' and become more impotent, precisely in proportion as the shape and constitution of things becomes more defined and settled.
So although their primary role is government of the world, their effectiveness and capability for direct action decreases as time passes, until eventually it is quite correct to say that they don't actually do much at all. A description of their behaviour in the Annals of Aman (HoME 10) during the First Spring of Arda is probably best to describe what is the most likely limit of their later activities too:
Now therefore the Valar were gathered upon Almaren and feasted and made merry, fearing no evil...
As for what their government of Arda actually entails, you can think of them as being like the back-room IT people who's job it is to make sure that your email stays working. You may not see what they're doing, you may not even know what they do, but the fact that you can get into your email means that they're doing their job.