Picture attached...

Okay what you've draw is tert-butanol, this doesn't undergo the sort of mechanism you've drawn (plus you need an arrow to break the C-O bond too), this would work fine for a primary alcohol such as methanol.

With t-butanol the oxygen picks up H+ with its lone pair, then this breaks away leaving C(CH3)⁺ (i.e. a carbocation) the Cl^- then attacks this carbocation forming the choloroalkane.

The mechanism you've drawn is called S_N2 but with t-butanol you have the S_N1 mechanism (this is for reference)

So... which alkanes undergo Sn1 and which undergo Sn2, and what determines this?

It's due to the stability of the carbocation and how easily the partially positive charged Carbon is attacked.

Carbocation stability: tertiary > secondary > primary

Likelihood of Sn2 increases this way ----->

What do you mean by primary, secondary and tertiary molecules/ions?

I've just been reading

http://www.chemhelper.com/sn1.html
and
http://www.chemhelper.com/sn2.html

They also keep making mention to this...