The modern evolutionary synthesis reconciled what Mendel had showed about the discrete nature of inheritance (or so it was thought), and what Darwin had insisted on was the major gradual, more or less continuous, nature of the variation in traits and their evolution.
A lot of things contributed to this reconciliation. Many experiments, largely stimulated by the 1900 rediscovery of Mendel's 1866 work, began to find mutations that were not grotesque, but had small effects, and were perfectly viable for the organism. Big bad mutations did still arise and had been obvious and easy to see, but now the majority of mutations seemed to be of the lesser type.
Quantitative traits like stature varied gradually rather than by discrete jumps (such as between yellow and green peas). Darwin's distant cousin Francis Galton had shown formally what had been obvious, that even for quantitative traits relatives resembled each other--indeed, they did so more than they might for traits with strong dominance (yellow peas did not resemble their green-pea ancestors in that trait). A variety of investigators, most notably the statistical geneticist RA Fisher (in 1918) recognized how the two types of inheritance could be brought under the same umbrella. If complex or quantitative traits were caused by the effects of many different genes, each making a small contribution, then the resemblance among relatives, the quantitative variation, and Mendelian inheritance were all consistent.
If that was so, then Darwinian gradual evolution was possible even with Mendelian inheritance as the basic fact of life. That's what the modern synthesis showed. It seemed wholly legitimate, indeed perhaps strikingly insightful, and it became the core theory of biology (and in many eyes, we think somewhat erroneously in ways similar to the adherence to Mendelism as the ground-state of inheritance, it still is). The formal theory of evolution or, to many, of life was the mathematical theory called population genetics. Well, Isaac Newton said something to the effect that if it can't be expressed mathematically it can't be a law of Nature, so we needed something real, not soft for biology!
It is strange for a central theory in a serious science to say nothing whatever about actual traits of the entities--organisms--that it purports to explain, but that's what population genetics does. It says only that variation is comprised of discrete Mendelian (that is, discretely transmitted) units of inheritance ('genes' has been the word for them), and that by chance and mainly by the systematic force of natural selection, changes in the frequency of genetic variants were responsible for variation among organisms and among species. Surprisingly, this theory became accepted even before there was any real theory of what a 'gene' is! That means it was such a blanket view, so mathematically rigorous, that it could be universally applied to entities that hadn't even been discovered. Reconciling this theory with actual traits has only come in recent decades under rubrics like EvoDevo (the evolution of development). Our book, Mermaid's Tale, deals with these issues extensively, and we try to show that there has been far too much stress in biology that rested on the Competition-among-clearcut-units-is-almost-everything worldview of the modern synthesis.
The dogma became that Mendelism and Darwinism are compatible.
But in a real sense, we think, this is fundamentally wrong. The modern synthesis essentially was possible only because Mendelian inheritance was itself wrong!