ॐ पूर्णमदः पूर्णमिदम् पूर्णात् पूर्णमुदच्यते | पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते || "Om Poornam Adah Poornam Idam, Poornaat Poornam Udachyate. Poornasya Poornam Aadaay, Poornam Evaa Vashishyate"

ॐ पूर्णमदः पूर्णमिदम् पूर्णात् पूर्णमुदच्यते |

पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ||

"Om Poornam Adah Poornam Idam Poornaat Poornam Udachyate

Poornasya Poornam Aadaay  Poornam Evaa Vashishyate" 

Simple Meaning:

'You are the fullness. There is fullness, here is fullness. From the fullness, the fullness is born. Remove the fullness from the fullness and the fullness alone remains.'

Implied Meaning:

That (Brahman) is infinite, and this (universe) is infinite. The infinite proceeds from the infinite. (Then) taking the infinitude of the infinite (universe), it remains as the infinite (Brahman) alone.

or

That (the invisible Brahman) is full, this (the visible Brahman) is full. This full (visible Brahman) proceeds from that full (invisible Brahman). On grasping the fullness of this full (visible Brahman) there is left that full (invisible Brahman)

Higher Level Meaning:

The Personality of Godhead is perfect and complete, and because He is completely perfect, all emanations from Him, such as this phenomenal world, are perfectly equipped as complete wholes. Whatever is produced of the complete whole is also complete in itself. Because He is the complete whole, even though so many complete units emanate from Him, He remains the complete balance.

Word by Word Meaning:

"Purnam" = complete

"Adah" =that

"Purnamadah" = "that" is complete.

"That" is referred to "Supreme Soul- Brahman"

(What we generally refer as God)

God is complete means what? Complete means He is self sufficient. He is Eternal. He IS always there. He does not need any external agency to support Him.

"Idam" = "this"

"Purnam Idam" = This is also Complete

"This" indicates the "Atman"

"ATMAN"= the individual Soul- which resides in every living being. "Soul" is complete means "self sufficient" like "Supreme Soul"

In other words the "Soul"

Which resides in our body is Identical with the "Supreme Soul"

"Purnat" = from (that) complete referring to "Supreme Soul"

"Purnam" = (this) complete referring to "Individual Soul"

"Udachyate" = has "come out" or emerged

This means from the "Supreme Soul" (which is complete by itself) this "Individual Soul" (Complete) has emerged.

This is the verbal meaning of the above. Now let us try to understand the same.

The Ocean is full of water. Thus referring to ocean we can say that (the ocean) is complete (Purnam)= full here.

From this, the small vessel is filled with water. Thus referring to the vessel also we can say this is Complete (Purnam) = full here.

Thus, we can say that the ocean is complete (full of water) and the vessel is also complete (i.e. full of water).

Thus, "Purnamadah Purnamidam" can be interpreted as Supreme Soul and the Individual Soul are both Complete i.e. whole and identical.

Purnat Purnam Udachyate meaning  from the Supreme Soul this Individual Soul is emerged.

Purnasya = of Complete referring to "Supreme Soul"

Purnam =Complete referring to "Individual Soul"

Aday =giving away to

(i.e. by giving away the completeness (wholeness)of "Supreme Soul" to "Individual Soul"

Purnam eva vashishyate only the Complete remains.

Now because a few vessels of water, if taken away from the Ocean, is the Ocean incomplete i.e. has it become less full of water? Or is it still full of water? It hardly matters to the Ocean (i.e., the level of water in Ocean) as to how much water is taken away.

Similarly from "Supreme self" individual self has emerged and is identical and even after taking away the soul, what remains is also the complete self.

Explanation of meaning with Example:

Thus as per this Shanti Mantra, the Atma(soul) which is there within ourselves is part of Brahman(God) because it has emerged from God.

Just as the mercury droplet when disturbed give us small droplets and when we bring them together, again merge into the big droplet. Thus our souls have also emerged from God and are destined to merge in God. Only difference is mercury droplet has definite quantity hence reduces in size when small droplets are formed. But Brahman being infinite being does not get reduced even after the soul is emerged. And even after the soul merges in to Brahman it remains the same.

But this is the ideal condition. Just as mercury droplets covered with dust do not merge fully into a big droplet, our souls covered with subtle cover of desires (to worldly things) do not merge with God after death but are born again and again. The aim of study of Upanishad is to cast off the dust of desire and let our souls merge with the God.