Voltage is the difference in potential between two wires. You can't say the "hot" wire has any sort of voltage by itself. Just as you can't say the "neutral" wire has any voltage by itself. The voltage is the measure of electrical potential between the two.
When you connect the light bulb between the two it provides a path for current to flow from high voltage to low voltage.
If you want to read about the specifics of this, Kirchoff's circuit laws (and especially Kirchoff's Voltage Law) are the place to start.
Before explaining current, we need to know what charge is, since current is the rate of flow of charge.
Charge is measured in coulombs. Each coulomb IS a large group of electrons: roughly 6.24 ˟ 10^18 of them.
The “rate of flow” of charge is simply charge/time and this calculation for a circuit gives you the number of coulombs that went past a point in a second. This is just what current is.
Resistance is a circuit’s resistance to current; it is, like you said, measured in ohms, but it is caused by the vibrations of atoms in a circuit's wire and components, which results in collisions with electrons, making charge passage difficult. This increases with an increase in temperature of the circuit, as the atoms of the circuit have more kinetic energy to vibrate with.
Voltage is the energy in joules per coulomb of electrons. This is shown though the equation E=QV where the ratio of Energy over charge= voltage. This is granted by the battery, which pushes coulombs of electrons, with what we call electromotive force. However when it is said that the potential difference across a component is X volts, it means that each coulomb is giving X joules of energy to that component.
Note: if an equation doesn’t make intuitive sense to you, chances are it is a complicated derivation, and to understand it you’ll have to learn its derivation.
Best Answer
Persistent currents
Persistent currents is the name given to the currents flowing without energy dissipation and therefore not requiring a voltage bias to be sustained. This is notably the currents in superconductors (already mentioned in the answer by @kruemi), but also currents in some mesoscopic devices, where the energy cannot be dissipated due to the size effects (e. g., lack of efficient coupling to phonon bath and/or restrictions on the electron momentum change). Note however that the currents on microscale generally do not follow the same rules as those for conventional circuits - in particular, the lumped-circuit description fails (see also my answer to Are voltages discrete when we zoom in enough?).
Diffusion currents, etc.
As less exotic options one could mention diffusion current, where the current is due to the difference in electron concentrations, rather than bias. This is however difficult to sustain for a long time, and it usually co-exist with conventional (drift) current, as, e.g., in pn junctions.
Thermoelectric and other phenomena
Onsager relations predict a number of phenomena where the current can arise from spatial variation in other thermodynamic parameters. The diffusion current above can be viewed as the result of a gradient in chemical potential. Another well-known case is thermoelectric phenomena. These are often expressed in terms of voltage at the terminals of a device, due to the accumulated charge, but in a bulk they are actually currents.