Where hydrogen belongs in the Periodic Table
This comes down to the fact that hydrogen has a $1s$ valence shell, capable of holding only two electrons.

In some sense, hydrogen is like the halogens, in that it can achieve the electronic configuration of a noble gas (namely, helium, which has a full $1s$ level) by gaining an electron. This reduction yields the hydride ion, $\ce{H-}$. It should be noted that hydride does not have the same stability as the halogen anions (e.g. hydrogen has an electron affinity of 0.75 eV vs. 3.40 eV for fluorine).

In another sense, hydrogen is like the alkali metals, as it only contains a single electron in its valence shell. Whilst the alkali metals can attain noble gas configurations by losing their valence electrons, it's a bit abstract to talk about hydrogen doing the same by losing its single electron to generate a proton ($\ce{H^{+}}$)(1). Nevertheless there is strong analogy between the electronic structure of hydrogen and the alkali metals which motivates hydrogen being often placed in this group.

Periodic table makers sometimes get away with duplicating hydrogen above the alkali metals and the halogens, or putting it in no group in particular.