There are many different approaches to combat ageing. Wikipedia is a good place to start learning about them.
If we are successful, humans theoretically will achieve biological immortality or near immortality. Specifics of how it can be done are outside of the scope of this question. So, let's just assume that people do not age anymore and stay in their peak physical form. However, they still can die:
- murder and suicide are possible;
- lethal poisoning by radiation, fast-acting poisons✽, etc. is possible;
- starvation will lead to death;
- auto-immune diseases and infections like Spanish flu will have higher fatality rates due to higher immune response.
Signs of age
Most of the effects of ageing will be psychological. Despite no signs of biological ageing (everyone looks young), older people will demonstrate more mature and conservative behaviour. It is important to note, though, that psychological development and maturation of non-ageing and long-lived/immortal humans will be different from ours. I will not be surprised if more stages appear, something like early middle adulthood and mid-adulthood.
When it comes to appearances, older people might prefer hair and clothing styles similar to their youth (but not necessarily wear them). They might also tend to dress 'age-appropriately'. Most of them will, probably, keep the same music and food preferences as in their youth (however, immortality might change neurological processes responsible for our preferences).
I would also suspect that older people will use slightly different language patterns. Their speech will include more archaisms. It is also quite possible that their social interactions will adhere to earlier norms (think about differences between you and your parents when it comes to social situations).
I also believe that humanity will experience a dramatic change in culture and society once ageing is defeated. Unfortunately, we can only speculate about its outcomes since there is no true frame of reference for immortality or permanent youth.
✽slow-acting poisons are most likely will be ineffective since boosted regeneration and immune response are required to stop ageing