Informally, a monad is a container of elements, notated as F[_], packed with 2 functions: flatMap (to transform this container) and unit (to create this container).

Common library examples include List[T], Set[T] and Option[T].

Formal definition

Monad M is a parametric type M[T] with two operations flatMap and unit, such as:

trait M[T] {
  def flatMap[U](f: T => M[U]): M[U]
}

def unit[T](x: T): M[T]

These functions must satisfy three laws:

1. Associativity: (m flatMap f) flatMap g = m flatMap (x => f(x) flatMap g)
   That is, if the sequence is unchanged you may apply the terms in any order. Thus, applying m to f, and then applying the result to g will yield the same result as applying f to g, and then applying m to that result.
2. Left unit: unit(x) flatMap f == f(x)
   That is, the unit monad of x flat-mapped across f is equivalent to applying f to x.
3. Right unit: m flatMap unit == m
   This is an 'identity': any monad flat-mapped against unit will return a monad equivalent to itself.

Example:

val m = List(1, 2, 3)
def unit(x: Int): List[Int] = List(x)
def f(x: Int): List[Int] = List(x * x)
def g(x: Int): List[Int] = List(x * x * x)
val x = 1

1. Associativity:

(m flatMap f).flatMap(g) == m.flatMap(x => f(x) flatMap g) //Boolean = true
//Left side:
List(1, 4, 9).flatMap(g) // List(1, 64, 729)
//Right side:
 m.flatMap(x => (x * x) * (x * x) * (x * x)) //List(1, 64, 729)

2. Left unit

unit(x).flatMap(x => f(x)) == f(x)
List(1).flatMap(x => x * x) == 1 * 1

3. Right unit

//m flatMap unit == m
m.flatMap(unit) == m
List(1, 2, 3).flatMap(x => List(x)) == List(1,2,3) //Boolean = true

Standard Collections are Monads

Most of the standard collections are monads (List[T], Option[T]), or monad-like (Either[T], Future[T]). These collections can be easily combined together within for comprehensions (which are an equivalent way of writing flatMap transformations):

val a = List(1, 2, 3)
val b = List(3, 4, 5)
for {
  i <- a
  j <- b
} yield(i * j)

The above is equivalent to:

a flatMap {
  i => b map {
    j => i * j
  }
}

Because a monad preserves the data structure and only acts on the elements within that structure, we can endless chain monadic datastructures, as shown here in a for-comprehension.