Why don't some symbols of the periodic table match the first or second letters of the elements?

Answer 1

The symbol for each element needs to have a different letter, or letters, to distinguish it from others. A single letter (in the Latin alphabet) is only enough for 26 different elements, and 118 elements are known so far, so two-letter codes are required for most elements.

If you mean, different to the element's name, that is (a) due to the requirement of having unique codes, and (b) because Latin names for some elements are different to the English names. For example, the Latin name for gold is aurum, so the symbol "Au" was chosen for gold.

Answer 2

The element's symbol sometimes matches the letters in the Latin word for it.

For example

Potassium's symbol K comes from its Latin name Kalium

Sodium's Na comes from Natrum

Antimony's Sb comes from Stibnium

Tungsten's W comes from Wulfram

Tin's Sn comes from Stanum

Lead's Pb comes from Plumbum

Silver's Ag comes from Argentium

Gold's Au comes from Aurum

Mercury's Hg comes from Hydragyrum

Others are named after what the founder(s) choose.

Answer 3

That's their abbreviation. Nitrogen, for example, is often abbreviated to "N" for simplicity.