Clarification of recourse to public funds and meaning

Hello there, I've read quite a few of the posts here and this may very well be a very dim question, but here goes.

I'm applying for a Fiance Visa then Spousal Visa, appreciate what having no recouorse means ie not available but exactly what does public funds mean and for how long is this? is it for 5 years until we make an Indefinite leave to remain?
I'll explain a bit more we're not spring chickens I look after my mum (85) and at some stage she's going to need a carer would my wife be able to apply for the carers allowance or is that a no go and I'd ahve to apply?

Just trying to get my facts right so I can mange her indoorses expectations.

Many thanks

What Country is your spouse native?