What is the precise definition of 'uniformly differentiable'?

Since I'm not familiar with manifold concept, let's restrict ourselves to functions with real domain.

Let $A\subset \mathbb{R}$ and $f:A\rightarrow \mathbb{R}^k$.

What is '$f$ is uniformly differentiable on $A$' referring to?

Let $f:[a,b] \to \mathbb{R}$. Differentiability means that the limit $$ \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$ (with the obvious modifications for $x = a,b$) exists, in which case we denote the limit as $f'(x)$. This definition can be rephrased as saying that there is a function $f':[a,b] \to \mathbb{R}$ which satisfies $$ \lim_{h \to 0} \left |\frac{f(x+h) - f(x) - hf'(x)}{h} \right| = 0. $$
The uniformity here means that we can approximate uniformly in $x$.

More precisely, given an $\epsilon > 0$ we may find a $\delta > 0$ so that whenever $0 < |h| < \delta$, then $$ \left|\frac{f(x+h) - f(x) - hf'(x)}{h}\right| < \epsilon. $$ It's easy to show that a differentible function is uniformly differentiable if and only if it's differentiable with a continuous derivative. I believe this is what Rudin has you prove.

Outside of Rudin's book, I don't know if I've ever heard the term "uniformly differentiable" used exactly, and a quick Google search seems to suggest that the term is primarily connected with that problem.