Well, it's worth delving a bit into the history of the periodic table of the elements to understand how it came to be.

Lavoisier is usually credited with the first systematic list of the elements, partially because he's generally credited as being the father of modern chemistry and is associated with the transition from alchemy to chemistry. His list, published in 1789, does have some errors: he included "light" and "caloric" (aka heat), and several metals are given in oxide form instead of pure form (e.g., magnesium, calcium, molybdenum, barium). There's no inherent order to the list.

As stoichiometry develops over the next few decades, chemists begin to discover the atomic weights of elements, and further establish relative ratios of elements in compounds, so we can identify that (say) magnesium oxide has magnesium and oxygen in a one-to-one ratio, whereas sodium oxide has a two-to-one ratio.

Parallel to this effort, it also becomes readily apparent that there are some groups of elements that behave very similarly--today, we'd call these the alkali metals (first column), alkaline earth (second column), chalcogens (third column from right; least used of the list), and the halogens (second column from right).

By the 1860s, these patterns and the atomic masses are starting to be combined into two-dimensional tables, taking a list of elements with increasing masses and chopping it off every so often to make a new row. Mendeleev wasn't the first to create this kind of table, but he was the first to recognize that some of the data was wrong [1], and most importantly, he discovered several periodic trends in the process that allowed him to predict the properties of unknown elements, with gallium and germanium the most ringing successes.

The periodic table we get in the 19th century does not look like our modern periodic table, though. It creates 8 groups of elements, with group VIII comprising three elements in a row (the columns headed by iron, cobalt, and nickel in the modern table). When noble gasses were discovered, they were combined into a Group 0. Not every row would have a group 0 and group VIII entry. Pretty much all of the tables from this era will list groups with their oxide/hydride valencies, which should give you an idea of how the groups are organized.

It's not until the early 20th century that chemists began to develop the theory of atomic numbers, and shortly thereafter electron configuration. This leads to the development of the periodic table dominated by increasing atomic number instead of increasing atomic weight, and also begins to recognize that the table isn't a simple table but has a more stair-step design, comprising what we would now call the s block (left two columns), p block (right 6 columns), d block (middle mass), and f block (the lanthanide and actinide series).

This development helps settle the issues with rare earth elements--the lanthanides don't make sense in view the old periodic table layout (there's too many of them!), although it also doesn't help that separating the lanthanides from one another is so difficult that several elements were later shown to be mixtures of different lanthanides. It really doesn't help organizing known elements into a table when your input data is wrong!

That said, if you pay careful attention to modern periodic tables, you will notice that there is one place where tables differ. Which elements are placed below yttrium? Is lanthanum and actinium in the main table? Or is lutetium and lawrencium in the main table? Or do you create a gap below yttrium and place both lanthanum and lutetium in the supplemental series at the bottom? Organizing the lanthanides is still an unsolved problem for chemists!

[1] At the time, the way of calculating atomic mass meant you had to make a guess at valency to compute the atomic mass. If the valency was wrong, then the atomic mass would be wildly off.