DULCE ET DECORUM EST are the first words of a Latin saying (taken from an ode by Horace). The words were often used as an oath and were frequently quoted by soldiers at the start of the First World War. They mean "It is sweet and right."
The full Latin saying can be found at the end of the poem: "Dulce et decorum est pro patria mori" These words are interpreted as "It is sweet and right to die for your country." In other words, it is a wonderful and great honour to fight and die for your country.
The irony lies in the speaker's declaration that this oath is "The old Lie." There is nothing sweet or good about war. There is nothing honorable in dying. The entire poem is about the ugliness of war and death in war. The last stanza says it all. The speaker reveals the truth about this sort of death to the unknowing eyes who haven't witnessed it and do not understand:
If in some smothering dreams you too could pace
Behind the wagon that we flung him in,
And watch the white eyes writhing in his face,
His hanging face, like a devil's sick of sin;
If you could hear, at every jolt, the blood
Come gargling from the froth-corrupted lungs,
Obscene as cancer, bitter as the cud
Of vile, incurable sores on innocent tongues,--
My friend, you would not tell with such high zest
To children ardent for some desperate glory,
The old Lie: Dulce et decorum est
Pro patria mori.
Below is a link to an in depth discussion of the poem "Dulce Et Decorum Est."
The poem is ironic because the title, which is a partial quotation from an ode by Horace, does not recognize the remaining three lines of the ode's stanza which are translated as follows: "Yet death chases after the soldier who runs,/ and it won't spare the cowardly back/ or the limbs, of peace-loving young men. . ."
This is no glorification of war. In fact it is the precise opposite. Thus one can see the irony of Owen's anti-war poem.