Random is a term used in mathematics (and less formally) to mean that there is no way to reliably predict an outcome (to know what will happen before it happens) or sense a pattern. Something that is chosen at random is not chosen for any conscious reason, and therefore thought to be purely by chance. An example of a random event is winning a lottery.
A computer can make lists of apparently random numbers. Humans are unable to do this, because the brain works in patterns. If someone is asked to keep saying "heads" or "tails" at random, a smart human observer or properly programmed computer might eventually be able to tell which one the person is likely to say next because the computer notices the patterns.
On a website such as English Wikipedia the user can click on "Random page" to get a random article. The chances of any one page appearing are exactly the same as for any other page.
Sometimes the word "random" is used more loosely. There are websites of random jokes, which just means: a variety of jokes about all sorts of things.
In recent years young people have started to use the word "random" even more loosely to describe anything which is rather strange or has no logic. Sentences such as "the mouldy cheese is escaping" or "I like pie and spam" might be described as "random", although this is not the proper dictionary or mathematical meaning of the word.