Question

When words begin with an alif why do they sometimes have a hamzah on them like, , and other times have a little squiggle like , and other times have nothing on top, as in . 

Answer

The alif with a hamzah written over it is really a hamzah sitting on an alif, not an alif.  Many books teach this as an alif, but that is not correct.  The second alif you referred to, the one that looks like: is called hamzah al-wasl.  That little mark over it is a miniature letter   
We only use this letter if we are starting on the word attached to it.  If you look in your Qur'an, you will notice that this hamzah al-wasl only appears at the beginning of a word that has a sukoon for its second pronounced letter. It maybe the second letter has a shaddah over it, but a shaddah represents two of the same letter, the first being the letter with a sukoon, the second being the letter with the vowel accompanying the shaddah mark, which looks like .

The vowel used on the hamzah al-wasl when starting depends on whether it is a verb, noun, or definite article attached.  The kind of vowel used to start words that begin with hamzah al-wasl will be covered in future tidbit lessons, insha' Allah

There are some copies of the Qur'an that don't write the hamzah al-wasl with the little over it, but instead just use the plain alif with nothing written over it. These same copies do not write hamzaat (plural of hamzah) sitting on an alif, instead the vowel  sign for the unwritten hamzah is over or under the alif.  When you see an alif with a vowel written over or under it, you can be sure that it is really a hamzah.  True alifs never have vowels accompanying it. It may be better to use the Madinah copy of the Qur'an which has the proper hamzah and hamzah al-wasl written in.