Let $X$ be a smooth projective variety over $\mathbf C$. How do we define an ample vector bundle $E$?
Do we just ask its determinant $\det $ to be ample?
Is it the same as saying that $f^\ast E$ is ample on $\mathbf P(E)$, where $f:\mathbf P(E)\to X$ is the projective bundle associated to $E$?
1) The answer to your first question is that by definition the rank $r$ vector bundle $E$ on $X$ is ample if the line bundle $\mathcal O_{\mathbb P(E)}(1)$ on $\mathbb P(E)$ is ample.
[ If $ r=1$ we apparently have two definitions for the amplitude of the line bundle $E$, but they clearly coincide since $\mathbb P(E)=X$ and $O_{\mathbb P(E)}(1)=E$.]
2) An important property is that a quotient bundle of an ample bundle is ample.
This allows to negatively answer your second question, whether a bundle whose determinant is ample is ample:
Take $X=\mathbb P^1$ and $E=\mathcal O(-1)\oplus \mathcal O(2)$.
Then $\det(E)=\mathcal O(1)$, which is an ample line bundle.
Nevertheless $E$ is not ample because one of its quotients is $\mathcal O(-1)$, which is certainly not ample.
[In the other direction it is true, however, that the determinant of an ample bundle is ample.]
3) Another important property is that the restriction $E|Y$ of an ample vector bundle on $X$ to any subvariety $Y\subset X$ is ample.
This immediately implies that for a vector bundle $E$ of rank $\gt 1$ the lifted bundle $f^*E$ on $\mathbb P(E)$ is never ample since its restriction to a fiber of $f$ is trivial, and thus certainly not ample.
This answers your third and last question, again negatively.
Edit
Although ample line bundles are an older concept, inextricably linked to the venerable notion of divisors, ample vector bundles of rank $\gt 1$ were introduced by Hartshorne only in 1966, in this article.
He gives another definition of "ample", but the one I chose (because it is closer to the notions in Jim's question) is equivalent by Proposition 3.2 of the article.
By the way, ample vector bundles of rank $\gt 1$ are not mentioned in Hartshorne's celebrated book and as an ironic consequence his generalized concept of "ample" is not as well known as it deserves...
