An ampere is a measure of how many electrons move past a point every second (though technically, it's movement of any charged particles, but for metal wires it's always electrons). 1 ampere = 6,241,510,000,000,000,000 electrons per second. A pipe with water moving through it could be measured in gallons per second. Same idea.
Watts are not just used in electronics. They're a measure of the rate at which energy is used or transferred. A stick of dynamite and a candle have similar amounts of stored chemical energy, but the dynamite releases it much faster than the candle, so the dynamite has a higher power output (for a shorter time). Likewise you could use two identical batteries in different ways. If one way uses more power, the battery will not last as long.
1 horsepower is about 750 watts, if you're familiar with cars. Just different ways to measure the same thing.
watts = volts * amps. So a 60 W bulb plugged into a 120 V socket will be drawing 1/2 an amp.
60 W = 120 V * 0.5 A
In AC circuits, the electrons are vibrating back and forth instead of going in a continuous loop. The frequency is just the number of vibrations per second. 50 Hz means they move back and forth 50 times per second.
It's important to understand the difference between current flow and energy flow, though. The actual electrons in a wire don't move very fast. In a DC circuit, the actual electron flow around the loop might be at the speed of molasses. The reason flipping a switch causes the light to turn on very quickly is because the energy flow is very fast. The energy is carried by waves in the electrons, not the electrons themselves. They are constantly repelling each other, so when you push some extra electrons onto one end of a wire, the others nearby jump away, which causes more near them to jump away, and so on, creating a wave of "push" that travels down the wire and then pushes on things at the other end. This wave travels from one end of the wire to the other at maybe 2/3 the speed of light, while the electrons themselves barely move.
Now, this does get confusing when you get to semiconductor theory, and I understand your issue. I can name one very important case. When working with charge pumps in the human body. Many places in biology the charge flow is positive. When taking a biomedical modeling class for EE we often had positive charge flow.
We can get crazier, what if you have cancer? There are many options, sometimes you pick radiation. Photon radiation exists, what about proton radiation? The amount of protons they are sending are measured in Amperes. Why? Positively charged particles per second(enjoy the pun).
The important part here your particle makes an issue of. If electrons were positively charged the issue would be swept under the rug by most people. The fact that they are negatively charged makes people think about what it really means.
If you really get down to the physics it is just a sign convention and is a menial problem. If you would like to assign them positive charge please do so, be internally consistent and do not publish anything and no one will be the wiser.
The most important thing is that if electrons were positively charged we would not have nearly as great of a name for the positron. I personally would not live in a world were the negatron is a particle.
Best Answer
You're imagining an open circuit to look like this:
A better analogy would be this:
The pipes in a circuit aren't surrounded by free space for the water to flow -- they are tunnelled through a rock. Where there is no pipe, there is just rock and the water does not flow.