Spontaneous mutations happen without any external influence. They are the result of natural errors that occur during DNA replication or cell division, and can happen in a variety of ways.

Errors during cell division: Mistakes can also happen when chromosomes are separated during mitosis or meiosis. Nondisjunction occurs when chromosomes fail to separate properly. This leads to gametes with too many or too few chromosomes, which is known as aneuploidy, and is what is responsible for conditions like Down syndrome (trisomy 21) or monosomy X (Turner syndrome).

Induced mutations are caused by exposure to mutagens - factors in the environment that increase the rate of mutation.

Ultraviolet (UV) radiation: UV light, especially UV-B, causes adjacent thymine bases to covalently bond with each other, forming thymine dimers. These dimers distort the DNA and can block replication or transcription unless repaired. 

Depurination occurs when a purine base (adenine or guanine) is accidentally removed from the DNA backbone, leaving behind a gap. This is caused by hydrolysis  from water (which can be a mutagen). Depyrimidation can also occur, but much more rarely.

Oxidizing agents readily transfer oxygen atoms to bases. One of the most common types of mutations that form from this is the oxidation of guanine into 8-oxoguanine, which binds with adenine instead of cytosine. Nitrous oxide, an oxidizing agent, can also cause deamination. This can turn cytosine into uracil, or adenine into hypoxanthine which pairs with cytosine.

There are molecules similar enough to the bases that they can be incorporated into DNA - these are known as base analogs. Some of these are prone to mispairing, such as 5-bromouracil, which is a thymine analog. It normally pairs with adenine, but frequently binds with guanine instead.

Used extensively in research (and chemical warfare) due to their highly reactive properties, alkylating agents add alkyl groups to bases. These are a small chain of carbon and hydrogen atoms. The most basic alkyl group is a methyl group, which contains one carbon and three hydrogens (–CH₃), but can include significantly more carbons and be fairly bulky side chains. These can alter base-pairing properties or result in deformation of the double helix structure.

Intercalating agents are flat, planar molecules that can insert themselves between base pairs of DNA. This deforms the double helix and can result in indel mutations.

Ionizing radiation, caused by things such as x-rays, results in the formation of highly reactive ions known as free radicals. These "steal" electrons from DNA, causing damage to it. These free radicals can form naturally from ordinary body processes as well (although at a lower rate) and this is why antioxidants are such an important part of one's diet - they have their electrons stolen rather than your own cells/DNA.