Oddities are the third type of numenera (next to cyphers and artifacts). As the name itself suggests, oddities are very specific and flavor-oriented element of the game. These items don't hold any practical value, and often are useless in combat. However, they are one of the elements building the world, and often reveal new details of the plot and the events around you. We can also use them as an additional (albeit small) source of gold. Very rarely, some of the oddities may be used in interactions with the world or other characters.
In short, oddity is a unique loot - an item that is picked up to be later sold in a shop. It means that you can freely collect them or sell them to merchants. By selling these items your party can become quite wealthy. The shins you obtain by selling them can be spent on new weapons, armor or cyphers. Some oddities can be also used (you must enter character equipment screen, select an item and check whether it is possible to use it) - this will increase the parameters of the main protagonist etc.
An example of an oddity would be e.g. an item creating a holograph of a human face that changes its expression during movement, or a plant that blooms only in total darkness.