A lightly edited version of a previous answer:

The late 1700s was a turning point in our understanding of extinctions. Before the late 1700s, in the Western European scientific tradition, the general understanding of a species was that it was in some way eternal.

After all, few of the assumptions that currently underlie biology were common in that science at that point. Few scientists had much of a sense of global scale, or of the extent to which animals in different areas of the world had fundamentally different evolutionary histories - after all, animals in different areas had been there as long as God put them there, and that was only thousands of years ago, not hundreds of millions (the genetic mechanisms of Darwinian evolution and the enormous age of the Earth not being well-understood until the 20th century).

Additionally, taxonomy as a discipline existed, but had not yet assumed its final form. What exactly a species is still has some blurred lines today, but it was much more poorly understood at the time. For example, it was assumed that giant tortoises around the world were all the same species, so nobody noticed when several species of giant tortoises went extinct. Similarly, in writing from 1697, 1774, and 1788, the extinct Irish Elk was considered the same as the North American Moose.

As a result, the extinction of the Dodo of Mauritius had also basically gone unnoticed when it happened; nobody really thought it would only exist on Mauritius and nowhere else. The Dodo seems to have been reported as being common by mariners on Mauritius up to the 1630s, but it seems sparser by the time the Dutch settled Mauritius in 1638. The last reliable report of a dodo seems to have occurred in 1662, by a shipwrecked sailor on an offshore island. The Dutch left Mauritius in 1710, and when the French recolonised Mauritius in 1721, they did not mention alive Dodos.

But because of all the assumptions I mentioned above about global scale and God's good will, nobody assumed that the Dodo was actually extinct. The Ashmolean Museum at Oxford University had a specimen in its collection until 1755, when - because it had become rotten - it was burned. The Ashmolean might not have done this, of course, if they realised that they would never find another alive Dodo.

In 1778, a Mauritian writer called Morel reported that the Dodo was locally extinct, but that it was probably a species like 'emus, cassowaries or even rheas' which was still found elsewhere. Many French scientists, on hearing about the mythical Dodo and the inability of Mauritians to find living specimens, assumed that the species was imaginary - real species don't go extinct!

In 1796, however, Georges Cuvier demonstrated that species in fact did go extinct, that the bones of mastodons and mammoths were not in fact the same bones as those found in modern day elephants but in fact entirely separate species. Extinction really had occurred. Cuvier was also the person who made Linnean taxonomy look much more like its modern form, with the addition of phyla to the taxonomic tree, and, importantly, the addition of extinct species to the taxonomy of life. Nonetheless, for a couple of decades after Cuvier, French scientists seem to have assumed that the Dodo was still imaginary.

Awareness of the extinction of the Dodo increased after the British invaded Mauritius in 1810. A writer called Prior in 1819 reported matter-of-factly that the Dodo was extinct. The popular image of the Dodo as an extinct animal dates from the 1830s - an author in Penny Magazine in 1833 used the Dodo to illustrate the agency of man in causing extinctions, and this seems to have caught the Britsih imagination.

By 1839, Darwin was using the Dodo as a comparison in his popular journals on the H.M.S. Beagle; Darwin warned that endangered species such as the (now-extinct) Falkland Islands Fox might go extinct: 'in all probability this fox will be classed with the Dodo, as an animal which has perished from the face of the earth'. Extant Dodo skeletons in European collections were re-examined in the 1840s, confirming that they were really different species to emus or pigeons. An 1849 article in a popular magazine, Blackwood's Edinburgh Magazine used the Dodo to highlight the dangers of the looming extinctions of animals like the kiwi and the Kakapo. Popular exhibits at the London Great Exhibition in 1851 and at the Museum of Natural History in 1860 further increased knowledge of the Dodo, as did the publication of Lewis Carroll's Alice's Adventures In Wonderland in 1865 (though even as late as 1867, the National Encyclopaedia was qualifying statements about the extinction of the Dodo, implying that it might still be found on other as-yet-undiscovered islands.)

Not coincidentally, Darwin's Origin Of Species in 1859 painted an now-widely-held view of species as being the result of many, many generations of mutations caused by random chance which have helped animals survive or reproduce. In the post Darwinian world, which saw nature in Malthusian terms, and saw animals as adapted to their environment, it stood to reason that an animal might actually only exist in a particular environment; there was no reason for God to put Dodos on more than one island.

So the period between 1796 and 1869 is a period of increasing awareness of the possibility and indeed fact of extinction. So, in the 1790s we have the first conceptions that it might be possible to hunt a species to extinction (thanks to Cuvier), and by 1850, the example of the Dodo was widely known. By 1878, comprehensive legislation was introduced in Mauritius giving protection to sea-birds in the area - clearly too late for the Dodo, but perhaps it may have saved other species from extinction.

Sources:

Gould, S. J. (1998). The Dodo In The Caucus Race. Leonardo's Mountain Of Clams And The Diet Of Worms. London: Vintage.

Turvey, S.T., & Cheke, A. S. (2008). Dead as a dodo: the fortuitous rise to fame of an extinction icon. Historical Biology, 20, 149-163.