As a TA myself, I find that most people have no idea - so perhaps I could explain...
In each class, there are always a few more 'challenging' children. Not necessarily naughty children, but children with various issues - such as emotional problems, or behavioural issues or some learning difficulties or specific disabilities that impact upon their learning.
A Teacher on their own would be spending a huge amount of class time simply managing behaviour without a TA and would find it difficult to help each individual child make progress. A good parent helper can really contribute here, but - as volunteers - cannot take responsibility nor should they feel compelled to have to. Step up the Teaching Assistant!
Teaching Assistants are generally well-trained and well-qualified, although many are simply 'nice' people who are good with children. What Teaching Assistants are not qualified to do is Teach.
One of the attractions of being a Teaching Assistant is that you don't have marking to take home with you, you have to Teach small groups - but generally deliver a lesson prepared and directed by your class Teacher and there's far less planning to do than a Teacher, which is great if you still have young children at home and not much time.
You don't get paid much, but it is wonderful - at the moment...
Academies are hiring Teaching Assistants now and expecting them to Teach. Spoiling an attractive job and forcing down wages for Teachers - beware!