A: The short answer to this is "many fewer rights than a tenant would" - but let's answer in more detail.
What is a lodger?
First, let's be clear about this. A lodger is someone who shares their landlord's main home, without having any exclusive access to any part of it. In practice, this means - for example - that the lodger should not be able to keep the landlord out of "their" room: if they can do that, say with a lock on the door, then they may be a tenant not a lodger, with all the extra rights that implies.
Sharing some living accomodation with your lodger helps the legal situation too: if this is the case, you won't need a court order if you ever need to evict your lodger. It doesn't need to be the whole property: for example, the landlord can have a private living room, but share the rest of the house with the lodger.
What rights does a lodger have?
The arrangement between a lodger and their landlord is much less formal than one between a tenant and a landlord. In most areas, it's really up to you what arrangements you want to make with your lodger. We'd always recommend having a written agreement with them, to cover not only things like when rent and bills should be paid, but notice periods when either party wants to end the agreement, and even practical considerations liking cleaning and overnight guests.
There's a lot to be said for communication. If you're sharing your own home with someone you don't really know well, it's important that both parties are able to be open about any issues that might arise. A formal "house meeting" once a month might sound excessive, but it beats leaving passive-aggressive notes on the fridge.
Setting expectations in advance is crucial too: if you don't want your lodger to bring overnight guests back, for example, then you need to spell that out clearly in advance. Likewise if you have an aversion to the smell of spicy food or expect them not to put packets of bacon in your vegetarian fridge. Most things can be solved by - well, acting like grown ups.
What if there are problems?
As you've asked the question about rights though, I'm thinking things may have deteriorated beyond this point. If you want a lodger to leave your property, you have to give them "reasonable notice": it's safest to do this in writing. Normally this is at least 28 days, but in cases where there are serious problems - aggressive, violent or illegal behaviour, for example - you can give a shorter notice period.
If your lodger refuses to leave, a landlord may be able to evict them without a court order (see the Protection from Eviction Act 1977 1977, section 3). If the property has been the landlord's main residence for the whole of the lodger's stay, having given written notice to quit at least 28 days ago, you can refuse to let your lodger into the property (for example, by changing the locks while they are out). Your lodger will still be entitled to the return of all their possessions. This really is a last resort though, and should only be used when polite verbal and written requests to leave have repeatedly been ignored.
If you want to read up on more information regarding lodgers and their landlords, Tessa Shepperson's series of posts for Lodger Landlord covers just about everything you'll need to know.
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And remember, if you advertise for a new lodger in the papershop window, Who knows what you'll end up with...