Brit here.
To answer this, let us first consider what the American President does: on the one hand there is a bunch of PR work and flag waving, on the other a bunch of keeping up with what's going on in the country,authorizing things and politicing.
In essence, the Queen handles most of the PR and flag waving, while the Prime Minister (a simplification in this case) handles most of the keeping up on things, authorizing and politicing.
Of course, the PM also does a bunch of flag waving and PR. The Queen also spends hours every day keeping up on what's going on in the country (and has done so for the past 60 years or so) so that she can discuss this with the PM in their weekly meetings. While the Queen may not have much recognized power anymore, a discrete comment of 'do you really think that is such a good idea?' from her will carry considerable weight.
Naturally, this is an over-simplification which glosses over things, and applies mostly to the UK rather than the other countries of which she is Queen.