The amp is what you connect your subwoofer to to get power from your car. The amp connects to the radio and your battery to get power to the subwoofer. You need an amp that can support the power levels of your subwoofer; i.e., if your subwoofer says that it has a MAX wattage of 1000 and an RMS rating of 500 watts, it means that your amp should be the same, at least for RMS. RMS is the constant power delivered to the subwoofer, while the max is the max amount of power delivered. If your subwoofer says that it runs at 2 ohms, your amp must be rated for 2 ohms or lower. An ohm is a measure of electrical resistance; the lower the ohms, the more watts you can get. DO NOT try to run a subwoofer that runs at 2 ohms on an amp that can only support 4 ohms. The amp WILL overheat and bad things will happen. Running a subwoofer that can run on 2 ohms at 4 ohms will only underpower it, not damage it. If you are going to run 2 subwoofers you'll either need an amp with 2 channels, or 2 amps. With 2-channel amps, the wattage that is shown is the TOTAL wattage. If you have a 2-channel amp that says it runs 2500 watts RMS, each subwoofer will receive 1250 watts constantly. The max watts of a sub aren't incredibly important. The RMS is what you should look at.
The capacitor stores electricity and is mainly a safety feature to protect the battery from being drained when your subwoofer hits really hard.
A really good subwoofer for under $300 would be this: http://www.pioneerelectronics.com/PUSA/Car/Subwoofers/Champion+Series+PRO/TS-W3002D2|D4
or this:http://www.pioneerelectronics.com/PUSA/Car/Subwoofers/Champion+Series+PRO/TS-W3002D2|D4
They are both Pioneer subwoofers which means that they produce sound with incredible quality and will last a long time.
With subwoofers, the more magnets better. Subwoofers usually only go up to triple-stacked magnets, but are most often double-stacked. The voice coil is what moves the cone (the front circle that you see) and the more the better. Subwoofers usually only have one or two voice coils. The part that allows the cone to move is called the surround, and they are pretty standard; if it's even a somewhat decent subwoofer, it'll have a much larger surround than smaller, non-subwoofer speakers.
MTX is decent, so almost anything you get from them will be fine.
My absolute favorite brand is Pioneer. Pioneer makes high-quality speakers, subwoofers, and amps that will last a long time. You can never go wrong with Pioneer.
Remember: Match the RMS of your subwoofer the the RMS of your amp, if you use 2 subwoofers, make sure you have enough power going to each subwoofer, and GET PIONEER SUBWOOFERS.