I'm confused about the two terms, when voltage is applied and across a certain element in the circuit.
What Ignacio said is the core of the answer, I hope I can help you out going a bit deeper.
Generally the only distinction between "applied voltage" and "voltage across" is how you are dealing with voltage itself:
- you apply a voltage to a bipole taking a voltage source and putting it in parallel with the dipole.
- you usually measure a voltage across some dipole, putting a voltmeter in parallel with it.
That's to answer your question. Now what if you apply a voltage generator? What would the voltage across it? The answer is: there is no answer. That is a limitation of the model we are using. Ignazio makes the useful example of a diode: you apply 5V but across it there's only something like 0.7V: that's because your voltage source has an internal resistance where the remaining 4.3V drops.
Remember that most of the times when you apply a voltage to a dipole, the voltage across it will be exactly what you are applying. The two wordings though does not mean the same thing at all.
Since this is at the top now, and I've read some others very good answers, and since the question is very basic I'd like to add two words about potential, a word that every answer uses. A potential is a scalar field associated with a vector field. This vector field must be conservative for the potential to exist, and for the electric field this is true only for electrostatic fields. When things start moving around no potential can be defined. I don't want to be the fussy physicist but a professor once throw a chalk at me for this imprecision (he was quite precise) so since this might be seen from young students I though this should be pointed out.
A voltage is always across two nodes, it is the difference between the electric potentials of these two nodes. They are strictly speaking always applied by something, but we speak about applying a voltage across two nodes when we set the potentials of those two nodes by connecting them to the outputs of a voltage source, which role is to make sure the voltage across those is fixed to a known value.
The voltage of a node is often a shorthand for the potential of that node with respect to the ground of the circuit (which, as a reminder, is only a node which has been arbitrarily associated to a 0V value).
Electric potential is often compared to height in the liquid analogy where the flow of water is electric current and rocks along its path, resistance.
Reminder: a node is a uniquely defined point of interest in the circuit (a pin, the intersection of several branches etc.).
The phrase "voltage across a circuit element" means precisely the potential difference between the terminals of the circuit element. One can measure this voltage with a meter.
The phrase "voltage applied to a circuit element" is less precise but I believe it means that one is driving the circuit element with a voltage source of some type and that the voltage across is, more less, fixed by this source.
The opposite of this would be the "voltage supplied by a circuit element" which would imply that the voltage across is generated by the circuit element, e.g., a battery, a charged capacitor, etc.
Applied voltage to a component is the actual voltage given to the component. Whereas voltage across a component is the voltage drop/voltage dissipated by the component. In both cases, voltage means a difference in electrical potential between two points. It is always between two points as it is only the difference between those two points that produce an electromotive force.
Now, applied voltage to a component and the voltage across a component may or may not be the same value. If you are applying 5 volts to a single resistor circuit, that resistor will get all of the 5 volts as voltage is conserved within a loop (KVL). If you now have 2 resistors in series of equal value, each resistor now gets 2.5 volts of the total 5 volts. Technically, in the latter case, a total of 2.5 volts is applied to the last resistor, and thus the voltage across it is 2.5 volts. However, there exists non-voltage driven components, meaning they're driven by current instead. A diode of any sort is a good example, where you may apply 5 volts to it, but the actual voltage drop may be something around 0.5 volts. In this case, the rest of the voltage is sent back to the source and power will be dissipated by the internal resistance of the source.
The right way to say it would be apply voltage across something - it sounds more precise. The voltage is a difference in potential between two points. So, when they say apply voltage, they omit the word across, assuming you know between/across which two points. Commonly, this phrase is used to say 'Apply voltage to a circuit', meaning provide power to a circuit, since you know where to connect two wires. Similarly, you can say apply voltage across the circuit, but it might sound somewhat redundant, but more precise. This phrase is rather used to specifically tell between/across which points the voltage should be applied or measured. In both cases it is meant the say the same thing, but might be misunderstood.