The 1X1 bricks (both square and round) were introduced in late 1954 or 1955 and used on the garage and small house sets and parts boxes put on sale at that time.
Together with the 1x2vs03 brick and the 1x2 macaroni, these were the first to have the LEGO logo engraved on the studs.
The main characteristic of the original version lies in the fact that the studs are not hollow. As a result, the stud surface often caves in at the time of manufacture, and shows a concavity (particularly notorious on some 1X1 white square bricks).
The square version has often a grayish tint, showing a manufacture affinity with the 1x1vs03 brick already mentioned. Seen from below, a circle may be visible under the stud but that (less than 0.1mm deep) is an ejector pin mark, and not a hollow. In case of doubt, particularly in the darker colours, probe inside the brick with a needle.
The mold pip is at the center of a basal edge.
The problem with the concavity of the stud surfaces was corrected when new molds were produced. The second version of the 1X1 bricks was probably introduced in 1956. Studs were now hollow (with a circa 3mm diameter hole).
The square version now has the pip near a corner (clearly visible as a protrusion in the Figure at left).
NOTE: I have several dozens of red 1x1 bricks, both square and round, apparently of this version but without the "LEGO" logo on the stud. These are clones not manufactured by LEGO.
The third versions of the 1x1 bricks were introduced circa 1960. These are distinguished by the smaller diameter of the stud hollows, correcting a possible fragility of the previous version.
The third versions of the 1X1 brick above are the last that will interest us. However, the figure at left illustrates a newer version of the round brick, introduced in the mid 1960s. It is characterized by a reduction of the diameter near the base (the newer brick is depicted on the left side of the Figure, compared with an old type on the right side).
At this time the logo had already changed to a taller type given inside a peripheral circle, which makes both brick types immediately recognizable.