As other answers have noted, a function is said to be convex (or "convex up"; I've never seen "concave up" before, although the meaning is obvious enough in context) if the line segment connecting any two points on its graph lies entirely above (or on) the graph between those points, and concave (or "convex down" / "concave down") if the line segment connecting any two points on its graph lies entirely below (or on) the graph between those points.
A rigorous algebraic definition, to complement this geometric description, is that a function $f$ is defined to be convex on a subset $S$ of its domain if and only if, for all $a,b \in S$ and all $t, s \in (0,1)$, $$t + s = 1 \implies t f(a) + s f(b) \ge f(ta + sb),$$ and concave if the opposite inequality holds (i.e. if $-f$ is convex). Further, $f$ is said to be strictly convex (or concave) if the corresponding inequality is strict.
(Note that the definition given above is often simplified by directly substituting $1 - t$ for $s$, but that somewhat obscures the underlying symmetry of the definition. The symmetric form also generalizes more readily to the various forms of Jensen's inequality.)
The connection between these two definitions is that any $x \in (a,b)$ can be written as the weighted average $x = ta + sb$, where $t + s = 1$ and both $t$ and $s$ are positive. Then $(x, f(x))$ is a point on the curve of $f$ at $x$, while $(x, y)$, where $y = t f(a) + s f(b)$, is the corresponding point on the straight line segment between the points $(a, f(a))$ and $(b, f(b))$.
Notably, this definition (in either its geometric or algebraic form) does not require the function $f$ to be differentiable or even continuous (although it can be shown that a function which is convex on an open interval must necessarily be continuous on the whole interval and differentiable at all but at most countably many points on it). Thus, it is more general than definitions based on derivatives and can be applied to more kinds of functions. For example, the function $f(x) = |x|$ is clearly convex on all of $\mathbb R$ according to this definition, even though it's not differentiable at $x = 0$. Thus, this definition, or something similar and equivalent, is usually taken as the fundamental definition of convexity, with more narrowly applicable ones like "a twice differentiable function is convex if its second derivative is non-negative" being proven as theorems.