The twelve links of dependent origination as I understand them are:

Ignorance --> Mental dispositions and volitional actions --->  Conditioned consciousness ---> Name and form (mind and body) ---> The six senses ---> Sense contact (sense impressions) ---> Feelings (pleasant or unpleasant) ---> Craving ---> Clinging ---> Becoming ---> Birth ---> Death

When it comes to reincarnation and how that works, in a very basic way. 

There is death. After death the mental dispositions, (tendencies of the mind) lead to volitional actions. These condition consciousness to seek out a new womb, a new body, a new state of becoming. This creates the body and mind, the khandhas. Which leads to the six senses: sight, sound, taste, smell, touch, thoughts and ideas. This in turn creates sense impressions. Which we feel as either pleasant, unpleasant, or neutral. These feelings lead to craving for more of the pleasant sensations and less of the unpleasant ones. We all tend to like pleasure and dislike pain. So we cling to what we like and want more of. We take on an identity. This is me, this is mine, this is self, the conceit 'I am'. This leads to becoming, which leads to birth, (and if you are a biological being of this Earth realm) old age, sickness, death and loss.

Rinse and repeat

 Upon death, ignorant of the mental dispositions and actions that lead to suffering (both mental and physical). Conditioned consciousness seeks rebirth, this leads to name and form (or body and mind)... and the cycle carries on. 

Because of ignorance it can go on indefinitely.

This is a very brief sketchy description,  Any gaps in my understanding are homework for me and the reader (-;

 It can be helpful to try and articulate one's understanding. I read somewhere that this is one of the ways we learn is through attempting to articulate our understanding of things. It doesn't have to be in writing. Can just be verbally out loud to oneself. This can be suprisingly helpful in checking one's understanding of a teaching and where there are gaps in one's knowledge, which points to what homework needs to be done.

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