(Not sure what an infused oil is? Confused about carrier oils? Read this first.)
There are two main ways to make an infused oil at home. For both methods, you will need a carrier oil, such as olive oil, some plant material (such as rose petals, cinnamon sticks, or dried bay leaves - or even a mix of a few different plants), a sterilized glass jar, and a strainer.
If you are planning to use fresh plants, start by thoroughly washing them, and patting them dry with a towel. You want to make sure your plants are totally dry before starting. It can be helpful to lightly bruise the fresh plants a bit, as this will help them to release their oils easier. If you're using a long stem of something - say, a sprig of rosemary - break it up into small pieces. If you're using dried plants, they're likely ready to use as they are, but they might also need to be broken up a bit.
For the first method you simply need to put some of your chosen plant into the jar. You do not need to stuff the jar full, and personally I usually do not fill the jar more than a quarter full, but it does depend a bit on the specific plant. Then fill the jar the rest of the way with oil, making sure that the plant material is completely covered. Then tightly cover the jar.
It will take several days for the herbs to really infuse into the oil. It's best to allow the jar to sit in a warm (but not hot) spot somewhere in the house. Each day you'll want to lightly agitate the oil a bit by gently tipping the jar upside down, then right side up again. Gently rolling it a bit also works, but you don't want to shake it. After about seven days open the jar and smell the oil. If it's as scented as you want, strain the plants from the oil, return the oil to the jar, and seal it.
If the oil is not as scented as you'd like, strain out the old plants, and simply repeat the process using fresh plants and the same oil.
Second Method / Heated Method
For the second method, you'll need a glass baking dish - do not use metal. Put your plants and the oil into the baking dish, and put it it into an over set on it's lowest setting. Some plants might only need to be set in the oven for perhaps an hour or two, while others may take longer. Check the oil often to see if the scent is where you'd like it to be, keeping in mind that it will be a little less scented once it's cooled.
Once it's done, allow the oil to fully cool, strain it, and store it in a tightly sealed sterilized glass jar.
Storage and Shelf Life
Infused oils should be stored in glass jars that are well sealed, and kept in a cool place that is out of direct light. Heat and light will both cause oil to break down and start to go rancid sooner. Some people like to keep some of the plant in the oil after it's made as a decoration. In many cases, this can also cause the oil to go rancid quicker.
How long the oil actually lasts depends a bit on plants used, method used, and
how they are stored. They can last a while, but really it's best to
make small quantities as needed. When your oil starts to smell funky, it's time to toss it and make a fresh batch.
"Oh no, botulism!" and Other Warnings
Botulism is an extremely nasty, if rare, disease. If you are planning to ingest your infused oils, please do a little extra research on the topic, and take a look at the extra steps you should take to avoid botulism and any other sort of foodborne illnesses or issues.
Likewise, please do a bit of research on the plants you intend to use for your infused oils. Even plants we usually consider to be non-toxic can cause some issues for people - as an example, oil infused with clove or cinnamon can be irritating to sensitive skin.
Beyond oil, you can also infuse vinegars, alcohol, and a host of other liquids... but that's for another post.