The first you see is a few little harmless-looking brown spots like these.
The spots just get bigger and bigger...
... until they start destroying whole leaves, then the stems too.
Finally the poor potato plant ends up looking something like this - every leaf shrivelled, every stem brown and sick-looking. You shouldn't let things get to this stage: when you first see the leaf-spots, remove the foliage completely, as rain will wash the fungal spores down through the soil and onto your potatoes otherwise (and you've never smelled anything bad until you've smelled a blighty potato).
The above photos were all taken on the same morning, of the same patch of potatoes ('Desiree', in case you're interested) so it just goes to show that blight comes on in stages, and some bits can be worse affected than others.
Anyway - so now all those stems have been cut off at ground level, bagged up like toxic waste (never compost them - the spores overwinter) and thrown away. I haven't quite dared lift the potatoes just yet: partly because I haven't had the time, but also because I detest the slimy mess of a blight-infected potato slightly more than I detest cleaning my downstairs loo. So both tend to get left for a long time in the spirit of procrastination (which in both cases generally just makes the problem worse).
One last thing - these are my 'Sarpo Mira' blight-resistant potatoes, on the same morning, growing just eight feet away:
This has deepened my admiration for this spud variety even more. It's a good roasting spud, though people say it falls apart if you boil it so it's steaming only. Anyway, from now on I'm making them a regular on my seed spuds order - when potatoes grow this well when blight is rampant all around, it's daft not to.