Monoatomic elements are those elements which have a single atom existing individually in nature.
For examples when you talk about hydrogen, oxygen you will write their chemical structure as H2 and O2.
This means element hydrogen exists as two atoms of hydrogen combined as H-H.
Similarly, oxygen exists as O-O again a combination of two atoms.
These are called as diatomic elements as they exist as two atoms in combined form in nature.
While you try to separate them as H and O instead of H2 and O2 they will be highly reactive and even explode.
Hence, by nature, they exist as a combination of two atoms due to instability in monoatom form.
Whereas monoatomic elements are those substances which have one atom existing freely in nature without in a chemical combination with another atom.
This means these elements have single atoms which are highly stable as such.
Noble gases as monatomic elements
All the noble gases called stable gases are monoatomic in nature. These include
Helium (He)
Neon (Ne)
Argon (Ar)
Krypton (Kr)
Xenon (Xe)
Radon (Rn)
Helium: It is the second element in the periodic table after hydrogen and is a highly stable inert gas. As seen in the picture below, it has two of its outer shells with electrons and does not electrons from other atoms for stability. H
Hence, the atom is highly stable and exists individually and not in pairs. Its atomic number is 2 and its mass number is 4.003.
Neon: This is the next noble gas of the list of stable gases. Like Helium, it also has a full valence outer shell with eight electrons.
The monoatomic elements exist as single atom as Ne with an atomic number of 10 and an atomic mass of 20.1797.
Argon (Ar): Another stable gas with 8 electrons in the outer shell. It is also a mono-atomic and is the third of noble gases. 
Krypton with atomic number 36 and mass number 83.798 is another noble gas to in the monoatomic form.
It has also fully filled valence electrons making it highly stable atom in-spite of being single.
Xenon: Is also another noble gas element with a single atom in structure.
It has atomic number 54 and atomic weight 131.293. It is also highly stable to completely occupied outer orbitals. The element is a single atom of Xe.
Radon (Rn): Radon is also a monoatomic structured element. It exists as a single atom of Rn.
The atomic number is 86 and atomic mass of 222.
Besides these inert gases, there are also heavy metals which can exist in monoatomic form with different physical and chemical properties when their lattice is disturbed. These include
Gold (79), Silver (47), Platinum (78), Palladium (46), Rhodium (45), Ruthenium (44), Osmium (76), Iridium (77) etc.
In nature metal atoms are monoatomic but they are bound by metallic bonds in a lattice such that more number of atoms of the same element are kept together. But there are no chemical bonds in between these atoms and hence can easily break up to form monoatoms by a certain force.
Even Hydrogen and other chemicals can also be found in the monoatomic form in space at specific temperatures.
