[cabaal]

In this picture I get that the H-Br bond is being broken to form an O-H bond and a Br radical. I don't understand why the furthest arrow to the left even exists. Why isn't it simply the below picture? Why is the lone electron from the oxygen being pushed off into free space?

[macman104]

All sigma bonds are comprised of two electrons.  So the single oxygen electron, and then one of the electrons from the sigma bond in HBr combine to form the new O-H sigma bond, leading to then the formaiton of the bromine radical.  So the notation in the top picture shows the combination of the oxygen radical and the electron from the H-Br bond, necessary to create the sigma bond.

[orgopete]

There aren't any "rules" for curved arrow usage. They seemed to have evolved to common usages, but if you were to search, you will find quite a bit of variance. I understand your question though. Based upon other curved arrow usage, the curved arrow showing electron movement and bond formation can be understood. However, for consistency, shouldn't you also have a similar arrow from the oxygen radical? If you did that, then the upper notation may seem more plausible.

If you want to see a survey I did with a class on curved arrow usage, see http://www.curvedarrowpress.com/chem/curvedarrowsurvey/curvedarrowsurvey.html
I chose several examples in which the curved arrows are deliberately ambiguous. (I argue ambiguity is not good pedagogy.)

[AWK]

as usal - wilipedia
http://en.wikipedia.org/wiki/Curved_arrow

[orgopete]

as usal - wilipedia
http://en.wikipedia.org/wiki/Curved_arrow

While I appreciate that one can find many descriptions of curved arrow usage, they have not been codified by the IUPAC, ACS or any other organizations that I am aware of. In that sense, there aren't any rules. If I wrote a set of rules, that would hardly be construed as a set of rules, except by me.

The reason I mention this is that in the linked Wikipedia article, the E1 elimination includes an incorrect usage of the curved arrows (by most chemists). While that example is not clearly ambiguous, it is the kind of example in which if the arrow head points between the bonds but not directly at the atom, one can wonder, is this a rearrangement or formation of a double bond.

[AWK]

Using of curved arrows for reaction mechanism is based on moving electrons (single or pair). There is not true statement of  orgopete that IUPAC or ACS did not disscussed this problem. See for example: Accounts of Chemical Research 22(10), 343-349, 1989).
Unfortumnately in the article from Wikipedia cited by me a basic citations are missing. Moreover examples of radical reactions are missing, but all examples of mechanism, even E1, are correct.
Chemists (between themselves) often use  incorrect or simplified mechanism notations that are understand correctly between them. I agree this should be forbidden in chemistry teaching.

http://chemistry.umeche.maine.edu/CHY252/Mechan.html (E1 is missing)

[orgopete]

@AWK, I would be happy to discuss the usage of curved arrows in another thread.

My original answer was to explain to the poster how I thought his curved arrows might have been interpreted. I have found great variety of curved arrow usage in textbooks, presentations, and the web. I am aware of IUPAC nomenclature and I am unaware of IUPAC curved arrow usage or guidelines or similar guidelines by the ACS. Without reviewing guidelines for authors, I don't think there are directions there, especially as curved arrows are uncommon.

Re: E1, see diagram. I interpret this as a hydride rearrangement. A similar use of rearrangement arrows,
http://en.wikipedia.org/wiki/Beckmann_rearrangement
http://en.wikipedia.org/wiki/Pinacol
an erroneous arrow,
http://en.wikipedia.org/wiki/Rearrangement_reaction

This is what I thought. The poster was using a curved arrow to show bond formation with the curved arrow ending on the atom. There are virtually two different curved arrow uses. A double barbed arrow ends on an atom for bond formation. However, with single barbed arrows, the arrow ends where the bond forms, therefore two arrows are required. If a single barbed arrow is used to show bond formation, then arguably a second arrow would be needed (blue below). This would be consistent, but unusual. I believed there is an inconsistency in curved arrow usage. In resonance structures, the arrows always end between atoms to indicate a change in bond order. Some authors maintain that symbolism and similarly end arrows between atoms to indicate a change in bond formation or bond order. While this is consistent, it is not common (except for radical reactions). I thought the poster was trying to understand this inconsistency.

[AWK]

E1 - good point. I attached file in which I show my solution for this ambiguousness. This precises information that only electrons are mowing.
The left side of the next scheme is, in my opinion, correct. The 'single barbed arrow' indicate move of 1 electron with attached substituent. As result 2 electrons form a bond (no additional arrow is needed).

In the mechanism of Beckmann and pinacol rearrangement some electrons are missing at the atoms taken part in it (O, N). This can be accepted between chemists, but in in encyclopedias or textbooks.
Article concerning rearrangement reactions shows more schematic drawing and can be accepted (except for teaching).

In these all cases from Wikipedia using of arrows is correct.

[orgopete]

What am I missing?