The periodic table is a systematic arrangement of the elements. They are arranged according to their atomic number, the number of protons (which in the case of atoms is also the number of electrons) of each atom.
Some of the properties of the elements can be predicted given their location in the periodic table. For instance, the reactivity of atoms is associated with the number of electrons, and therefore, reactivity of some atoms can be predicted based on their location on the periodic table. To cite a few examples, elements belonging to the second column (Group 2) usually form +2 ions such as Mg2+ and Ca2+, and hence, when reacted with, say OH-, forms Mg(OH)2 and Ca(OH)2. This being said, elements belonging to the same group have the same number of valence electrons, the outermost electrons involved in chemical reactions. All elements in Group 18, as another example, are noble gases, and are hence mostly inert - these elements all have a complete octet (8 valence electrons), and are therefore, already stable on their own. This is probably the most useful property that can be predicted from the location of an element in the periodic table. However, there's also what is known as the Law of Triads established by Dobereiner, in which it was shown that some elements can be grouped in threes. For instance, in Group 1, Lithium, Sodium, and Potassium can be classified as a triad of reactive metals. In this particular triad, the atomic weight of the middle element (sodium in this case) is the average of the 1st (Li) and 3rd (K) element. There are a few other of these triads, and they can occur either in a column or group as illustrated here, or in rows (try carbon, nitrogen, and oxygen).
There are also trends in the periodic table - ionization energy, which increases from left to right, for instance, due to number of electrons and protons in a given atom, or atomic size which increases from top to bottom due to the 'size' of orbitals. These arise from the number of protons (and electrons), which, again, is the basis for the arrangement of the periodic table.
The chemical name of an element cannot be predicted from the periodic table. Chemical names are given to elements as they are discovered. Sometimes these are written in periodic tables, but often times the chemical symbol - or the shorthand name of that particular element - is what is written in the periodic table. Usually, it is easy to determine (or memorize) the names and symbols of elements - for instance, C stands for Carbon, Ne for Neon, F for Fluorine. Other times, these are not so obvious - as in Cs for Cesium, or Pt for Platinum. Still, other cases are more difficult, especially when symbols are based off Latin names instead of the common names - as in Ag for Silver (Argentum) or W for Tungsten (Wolfram). The periodic table is a way to organize these elements, and is not a way to predict chemical name and symbol.
Finally, while it might appear that some of the elements in the lower portion of the periodic table (the UU series, for instance) are more recently discovered than the upper portion, this is not actually the case. Take for example, Uuo and Uup (elements, 118 and 115, respectively). Uuo was discovered in 1999 while Uup in 2004. Again, the periodic table is arranged in order of atomic number and not in order of discovery.
In brief, the periodic table is a systematic arrangement of elements, and this arrangement is based off their atomic numbers. Elements belonging to the same group have, usually, similar properties, and there are usually trends that can be extrapolated from their sequence in the rows and columns (so-called periodic trends). This being said, certain generalizations can be made and some chemical properties predicted.