The picture shows the plant that Mendel has become famous for, the common pea Pisum sativum, but it also cleverly shows an illustration of the "laws" that he discovered. We can see the traits green (dominant) and yellow (recessive) color be passed on through two generations in the predictable proportions dictated by Mendelian inheritance. This is good old basic school biology, and most of us must be familiar with Mendel from that context. The Guardian also commemorates the day and provides a little bit of background.
Much has been said, and probably remains to be said, about Mendel's role as "father of genetics" and whether or not he predicted the existence of genes. I always think it's fun to go back to the original source of things, so here we have the relevant piece from Mendel's 1866 paper Versuche über Pflanzen-Hybriden or Experiments in Plant Hybridization, which was first read to the Natural History Society in Brünn (now Brno in the Czech Republic) the year before. I recommend this site for insightful comments on and facts about Mendel's original paper.
With Pisum it was shown by experiment that the hybrids form egg and pollen cells of different kinds, and that herein lies the reason of the variability of their offspring. In other hybrids, likewise, whose offspring behave similarly we may assume a like cause; for those, on the other hand, which remain constant the assumption appears justifiable that their reproductive cells are all alike and agree with the foundation-cell of the hybrid. In the opinion of renowned physiologists, for the purpose of propagation one pollen cell and one egg cells unite in Phanerogams* into a single cell, which is capable by assimilation and formation of new cells to become an independent organism. This development follows a constant law, which is founded on the material composition and arrangement of the elements which meet in the cell in a vivifying union. If the reproductive cells be of the same kind and agree with the foundation cell of the mother plant, then the development of the new individual will follow the same law which rules the mother plant. If it chance that an egg cell unites with a dissimilar pollen cell, we must then assume that between those elements of both cells, which determine opposite characters some sort of compromise is effected. The resulting compound cell becomes the foundation of the hybrid organism the development of which necessarily follows a different scheme from that obtaining in each of the two original species. If the compromise be taken to be a complete one, in the sense, namely, that the hybrid embryo is formed from two similar cells, in which the differences are entirely and permanently accommodated together, the further result follows that the hybrids, like any other stable plant species, reproduce themselves truly in their offspring. The reproductive cells which are formed in their seed vessels and anthers are of one kind, and agree with the fundamental compound cell.
Mendel is clearly talking about "elements" and the arrangement of these elements within the gametes, the egg and pollen cells, and therefore of the cell that gives rise to the whole organism. He starts out by pointing out that observable variation must result from variation in the gametes: not all egg cells are the same, and not all pollen cells are the same. The cause of the variation is then the difference in the "material composition and arrangement of the elements" carried in the gametes. When the egg and pollen cells unite, these differing elements must also unite within the resulting "foundation cell" in a way that has to be constant from generation to generation. There must be a "constant law". If the two parents are the same, it follows that the offspring will be the same as its parents: they "reproduce themselves truly in their offspring". But what he's really describing is that must happen if the parents are different; then it follows the the offspring will be different from either of the two parents, it will have "a different scheme".
Now comes the part that is really brilliant.
With regard to those hybrids whose progeny is variable we may perhaps assume that between the differentiating elements of the egg and pollen cells there also occurs a compromise, in so far that the formation of a cell as the foundation of the hybrid becomes possible; but, nevertheless, the arrangement between the conflicting elements is only temporary and does not endure throughout the life of the hybrid plant. Since in the habit of the plant no changes are perceptible during the whole period of vegetation, we must further assume that it is only possible for the differentiating elements to liberate themselves from the enforced union when the fertilizing cells are developed. In the formation of these cells all existing elements participate in an entirely free and equal arrangement, by which it is only the differentiating ones which mutually separate themselves. In this way the production would be rendered possible of as many sorts of egg and pollen cells as there are combinations possible of the formative elements.
Once these differing "elements" have settled in the offspring cell, they have formed "an arrangement" between them, a "differing scheme", the whole arrangement isn't passed on to the next generation. Rather, the "elements" "liberate themselves from the enforced union", once again migrate into differing gametes, and combine independently of each other, in new ways, in the generation that follows. Mendel's "elements", just as genes, are distinct and particulate. This has become known as Mendel's laws of inheritance: The law of segregation and the law of independent assortment.
So here Mendel is laying the whole foundation for meiosis, genetic variation and heredity, entirely from the basis of keen observation and meticulous and numerous hands-on experimentation. And, we must not forget, a great deal of luck. I think this insight is more interesting and more valuable than discussing how well he predicted the existence of genes as we know them today. I prefer to think that, in a way, Mendel left a "gene-shaped" hole through which the right peg could be put.