When an elliptic curve is said to be "over a field $K$", we mean that the curve is the set of "points" $(x,y)$, where both $x$ and $y$ are elements of $K$, such that the curve equation is fulfilled. The curve equation may have various representation, but a typical one is $y^2 = x^3 + ax + b$ for two constants $a$ and $b$ which are also elements of $K$. An additional formal point, the "point at infinity" (denoted $\mathcal{O}$), which has no $x$ and $y$ coordinates, is also part of the curve.
In cryptography we are mostly interested in finite fields. A finite field is traditionally denoted $\mathbb{F}_q$ for some integer $q$, which is the size (cardinality) of the field; it so happens that when two finite fields have the same number of elements, they are isomorphic to each other, which more or less means that they have the same structure; morally, both are "the same" field, which is why we can afford to say "the field $\mathbb{F}_q$. There exists a finite field $\mathbb{F}_q$ if and only if $q = p^f$ for a prime $p$ and an integer $f\geq 1$. When $f = 1$, i.e. the field cardinality is a prime integer $p$, then the field $\mathbb{F}_p$ is (isomorphic to) the set of integers modulo $p$.
In a finite field $\mathbb{F}_q$, there are $q^2$ pairs $(x,y)$, so the curve cannot contain more than $q^2+1$ points (including the point at infinity), which means that the curve is also a finite set of points. The cardinality of the curve is its size: the number of points it contains (again, including the point at infinity). Actually, the curve cardinality is close to $q$: if we define $t$ such that the cardinality of the curve is equal to $q+1-t$, then it can be proven that $|t| \leq 2\sqrt{q}$ (that's Hasse's theorem).
Roughly speaking, for about half of the $q$ possible values $x$ in the field $\mathbb{F}_q$, there can exist a curve point $(x,y)$, and when there is one, there usually are two. The "two" and the "half" mostly cancel out, and you end up with about $q$ points on the curve.
For practical usage of elliptic curves, we use a conventional generator which is a point $G$ on the curve. That point has an order: the order $n$ of the point $G$ is the smallest integer $n \geq 1$ such that $nG = \mathcal{O}$. There are $n$ points $kG$, for all integers $k$ from $0$ to $n-1$, and these are, collectively, a subgroup of the curve. This implies that $n$ is a divisor of the curve cardinality. We usually look for curves and generators such that $n$ is a big enough prime; if possible, we arrange for the curve cardinality to be itself prime, and then $G$ generates the whole curve (in particular, any point on the curve, except $\mathcal{O}$, can be used as generator).
For real-world encryption on the curve, well, we do not really encrypt. We use Diffie-Hellman. The DH algorithm produces a shared secret value, which happens to be a curve point. The point selection is basically random, but a shared random value is good enough because it can be used as key (after a suitable hashing) for symmetric encryption. There is a name for that combination: ECIES. But some standards just do the DH and then use their own symmetric encryption system (typical of SSL/TLS with the "ECDHE" cipher suites).
ElGamal could also be used, but it would require mapping, in a reversible way, a "message" $m$ into a curve point. This would be interpreting the message as the coordinate $x$ (in $\mathbb{F}_q$) of a curve point, and computing a matching $y$. However, only about half of the $x$ values can be the first coordinate of a curve point, so some "variability" would be needed (some bits which can be adjusted until a possible $x$ is reached), or some other trick (e.g. not using one curve but two curves which complement each other). Since the size of the message $m$ would be severely limited (no bigger than $q$), this is hardly worthwhile, and nobody does that. People use Diffie-Hellman.
The guide to elliptic curve cryptography is a good reading for a practical view on how things are done. The chapter on finite fields can be downloaded for free, so I warmly encourage you to read it first.
