Most car speakers are "coaxials" (2-way) or "triaxials" (3-way). There are also "component" speakers (AKA: "separates") which can be had, which are usually higher quality, more expensive, and usually require more power to drive them properly.
Coaxial and triaxial speakers are simply a 2 or 3 driver speaker system integrated into a single "speaker" package. They usually have a built-in simple passive crossover to divide and send the high frequencies to the tweeter and the lower frequencies to the midrange driver. Some of them don't have a crossover at all, and rely on natural rolloff frequencies to accomplish the same thing, albeit in a less precise manner.
Component speakers have separate drivers and a separate crossover, which is higher quality than the integrated crossovers in a coaxial speaker.
They all work the same way though. They have a positive and negative terminal, and that is all you need to hook up as far as input from the amplifier is concerned. With component speakers you also have to connect the crossover to the tweeter and midrange driver, but I doubt you will be using component speakers.
As far as impedance goes, check to see what the impedance of the original speakers are. Ideally you want your replacement speakers to be the same nominal impedance. Car speakers are usually 4Ω, while home speakers are usually 8Ω. I don't know what PC speakers usually are.
Using speakers that are of a lower impedance than your amplifier is rated for is hard on a typical amplifier, because it doubles the current flow each time the impedance is halved, given the same amplifier output voltage. However, to a point, this can be countered by using a reasonable volume level (because the volume knob controls output voltage). For example, using 4Ω speakers with an amplifier that is rated for 8Ω is usually fine, as long as you don't get crazy with the volume. I wouldn't suggest connecting something ridiculous to your amplifier like a .5Ω load however (e.g., eight 4Ω speakers wired in parallel to a single channel).
Going in the other direction is perfectly safe; i.e., using speakers that are of a higher impedance than your amplifier is rated for. This actually makes life easier for your amplifier, though you will get less power from it (half the power at a given amplifier output voltage each time you double the speaker impedance).
Don't worry about the wattage that the speakers are rated for. The Logitech x230 amplifier only sends 6 watts x 2 to the mids and highs. Any car speakers can handle that; even cheap OEM ones from like an '81 Chevette.