As you call it an "exhaust" I assume it's an extracting type rather than a filtered recirculating type.
It's a good idea to run it:
- Whenever you're frying (it handles grease droplets in the air as well as smoke)
- When you have the potential to produce smoke (so not generally when cooking in the oven, but sometimes, if things might drip or spatter).
- When you're cooking something that might make the house smell for longer than you'd like.
Generally speaking you should use it in anticipation of smoke/grease/odours, though using it reactively for toaster smoke is also a good idea.
Boiling is another matter. If the house gets damp, running the fan when boiling will reduce that (as of course will boiling with a lid on). Some houses end up rather dry (e.g. with forced air heating), in which case it's probably not worth turning the fan on. In summer I'd rather open the window.
It's worth thinking of where the air comes in to replace that sucked out by the fan. I have a rather open plan living area, and open a little ventilation at the far end from the kitchen, so the fresh air comes through the sitting and dining areas to the kitchen. But here in the UK we have "trickle vents" over many windows that are designed to deliver fresh air without you feeling a draught unless you're very close.