My 15 year old dog started leaning towards her left side yesterday. When she does this she becomes rigid and shakes a bit. I have also seen her become rigid and shake when a tiny bit when she is lying down (she looks to be in an awkward position lying down too. Today she won't walk up or down the stairs, she wouldn't bend down to eat her food so I had to feed her myself and she had difficulty moving lowering herself to the floor to lie down.
She went to the vet yesterday and they said she has dementia( she is a bit odd sometimes e.g. Just stands in the middle of the room staring at nothing) and possibly arthritis. What is happening?

By "leaning" do you mean the whole dog? Or is her head tilting to the left?
Does she walk in circles to the left?
Dementia or not she shouldn't be rigid and shaking.
At 15 it could be anything from vestibular disease to more serious issues.
As far as the reluctance to do stairs and the trouble laying down, yes likely arthritis, just my guess.
But you may need a second opinion regarding the rigidity and shaking.

Most dogs that are 15 develop some sort of problem as this is when the dog gets old.it sometimes depends on the type of breed as different breeds have different problems.when I was younger my grandma had a 15 year old lab and it started to get very stiff it couldn't eat it couldn't sleep it couldn't much do anything what we had to do was put him down but I don't think that's the case with your dog.when she is standing up from lying down does she yelp or show any signs of pain if it is her head that is leaning to the left one of my guessues would be an inner ear problem but it is useally something to do with the ears.please keep in mind that your dog is 15 if you used to take her out to the park and stuff I would recommend to keep her in the house and let her rest.but then as ladysam stated it could be something more seriouss.