Essential oils contain a strong essence of the plant they were derived from. Essential oils are generally very concentrated, and usually no more than a few drops at used at a time.
Essential oils are frequently made though a distillation process, but there are some other methods used as well, such as using a press to extract the oil from the plant. Essential oils are not made by simply steeping plant material in oil, that results in an infused oil.
When the distillation method is used, it produces both the essential oil, as well as what may be called an essential water. For example, if you were to use roses in this process, you would end up with both rose essential oil, as well as rose water, which is often used in cooking, baking, or as a beauty product. The essential water is not always kept, as depending on the plant used it may have an unpleasant smell or taste.
It is difficult to make essential oils at home, as generally one will need special equipment, and it requires a lot of plant material to get a good amount of oil for your efforts. One example I've seen given is that one would need 5000 pounds of roses to make a single pound of the essential oil. Many essential oils can be quite expensive because of this - but again, since a very little can go a long way, a single bottle will usually last a long while when stored properly. Essential oils should be stored in a tightly sealed (ideally dark colored) glass bottle, and left in a cool, dark place. Too much exposure to air, sunlight, heat, or extreme temperature changes will start to break the oil down.
Infused oils are much easier to make at home than essential oils. To make an infused oil one simply needs to let some plant material steep in a base oil, such as olive oil, long enough that the scent and flavor from the plant can transfer into the oil. For more information on how to make and store infused oils, please see this post.
Carrier oils are used to dilute essential oils, and may also be used to make infused oils. Many carrier oils have very little scent of their own, which make them good bases to work with. There are many different kinds of carrier oils, but some of the more popular choices include olive oil, almond oil, canola oil, walnut oil, peanut oil, or apricot kernel oil.
Which carrier oil you use will come down to personal preference most of the time. For making infused oils that will be ingested, olive oil is probably one of the most popular choices. Chances are you've seen, or eaten, flavored olive oils in the past. For other uses, some carrier oils may be a bit thicker than others, have slightly different scents, colors, prices, and so on. Some are readily available in the grocery store, while others may be harder to find. Look around, experiment a bit with what's out there, and see which work best for your needs.
I'm often asked if mineral oil, or baby oil, can be used as a carrier oil. Obviously if you're going to be ingesting your oil, you probably don't want to be using mineral oil. If you were making something like an anointing oil for use on your skin, mineral oil might be an okay choice - however, mineral oil doesn't absorb into the skin very well. So while it can be used in some cases, one of the plant based carrier oils will usually make a better choice.