Electric current is the steady flow of charge. Electrons and protons are jiggling around all the time due to thermal noise, but it's only considered current when a bunch of charges all move in a more-or-less single direction for a while. More specifically, current is the amount of charge that flows past a particular point (in a wire, for instance) per second. The SI unit of current is the ampere (A):

$$1\u{A}=1\u{C\over s}$$

We will see in   TBD   that current is proportional to the number of charge carriers in the wire, and also to their speed.

Electrons move in the opposite direction of the indicated current.

In a typical metal wire, the charge carriers are electrons, and any current that flows through the wire is due to negative charge. However, to avoid the complication of additional minus signs in our calculations, current is said to flow in the direction positive charge would move. Thus, if a wire is said to carry 1 A of current to the right, that means that the electrons in the wire are actually moving to the left.

The net charge of each section of wire remains zero.

We know that a coulomb are large charges, so one might suspect that an ampere of current— one coulomb of charge flying by every second!— would be huge. But that's not so: while an ampere is perhaps a bit larger than a typical current, it's not unusually so. Why not? While a coulomb of electrons may flow by every second, their charge is cancelled out by an equal number of protons which remain stationary in the wire. As electrons leave one section of the wire, new electrons enter that same section, and the net charge of the section remains zero.

charge building up inside a box

Suppose this didn't happen. Imagine a box with two wires sticking out of it. Four amperes flow into the box through one wire, and 3A flow out of the box through the other wire: in other words, four coulombs of charge are entering the box every second, but only three of those coulombs are leaving: one coulomb of charge (per second) must be remaining inside the box. The box is building up a large positive charge, very quickly! If I try to push new current into the box, those positive charges are going to be repelled by the charge on the box, and so will slow down: the incoming current decreases. Meanwhile, positive charge will try to escape the box through the other wire, increasing the outgoing current. Pretty soon, the currents will have adjusted themselves until you have 3.5A in and 3.5A out. While it's possible to have a situation as described above---more current in than out, or vice-versa--- it's a situation that won't last for long. If the currents into and out of a box maintain some constant value over a long time (what we call steady current), then the current in is equal to the current out. This is called current conservation.

Three wires are joined together at a point (called a junction), as shown. The current in two of those wires is given. What is the current in the third wire?
A) 4A up B) 2A up C) 2A down D) 4A down

3A flows out of the junction, but only 1A flows inward. We need another 2A to flow into the junction, and so the current in the third wire is 2A upward.