How can an object's instantaneous speed be zero and it's instantaneous acceleration be nonzero? I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation.
 A: Suppose you throw a ball upwards at some velocity $v$. When you catch it again it's traveling downwards at (ignoring air resistance) a velocity of $-v$. So somewhere in between throwing and catching the ball it must have been stationary for a moment i.e. it's instantaneous velocity was zero. Obviously this was at the top of its travel.
When you throw the ball it immediately starts being accelerated downwards by the Earth's gravity, so it has a constant acceleration downwards of $-9.81ms^{-2}$ (the acceleration is negative because it's reducing the velocity of the ball).
So this is an example of how there can be a non-zero acceleration (of $-9.81ms^{-2}$) but there can be a moment when the ball's instantaneous velocity is zero.
A: Small addition to John Rennie answer.
In fact, if you hold a ball in your hand and will release it, then the ball starts its motion with non zero acceleration but zero speed. 
I think it is not trivial fact. There is an interesting story about it in the history of science. Galileo Galilei spent a lot of time to understand how a body can start its motion with zero velocity. For his time it sounded like an absurd and nobody believed him. In his book «Dialogues Concerning Two New Sciences», he convinced himself (and everybody else) by the following remarkable explanations why $v(t)$ can be zero for $t=0$:

You say the experiment appears to show that immediately after a heavy
  body starts from rest it acquires a very considerable speed: and I say
  that the same experiment makes clear the fact that the initial motions
  of a falling body, no matter how heavy, are very slow and gentle.
  Place a heavy body upon a yielding material, and leave it there
  without any pressure except that owing to its own weight; it is clear
  that if one lifts this body a cubit or two and allows it to fall upon
  the same material, it will, with this impulse, exert a new and greater
  pressure than that caused by its mere weight; and this effect is
  brought about by the [weight of the] falling body together with the
  velocity acquired during the fall, an effect which will be greater and
  greater according to the height of the fall, that is according as the
  velocity of the falling body becomes greater. From the quality and
  intensity of the blow we are thus enabled to accurately estimate the
  speed of a falling body. But tell me, gentlemen, is it not true that
  if a block be allowed to fall upon a stake from a height of four
  cubits and drives it into the earth, say, four finger-breadths, that
  coming from a height of two cubits it will drive the stake a much less
  distance, and from the height of one cubit a still less distance; and
  finally if the block be lifted only one finger-breadth how much more
  will it accomplish than if merely laid on top of the stake without
  percussion? Certainly very little. If it be lifted only the thickness
  of a leaf, the effect will be altogether imperceptible.

The second funny story was about equation of motion: since $v=0$ in the beginning of motion, Galilei assumed that the speed of free falling body should be proportional to the distance passed: $v(t)=a\,l(t)$, where $a$ is a some constant. Later he proved that such motion is simply impossible for the initial condition $l(0)=0$. Then he found the correct equation $v(t)=a\,t$.
Later John Napier consider Galileo's equation $v(t)=a\,l(t)$ for $l(0)\neq 0$ and thus he discover logarithmic function!
A: Throwing a ball in the air and having it come back down is a basic concept of why there is non zero acceleration at a zero velocity.  You throw a pencil up in the air and at the apex of the projectile,  the velocity will always be zero but because of acceleration due to gravity (9.81m/s^2),  the acceleration will never be zero. 
