I see a lot of people conflating the various terms for "smallish mobile computing device intended mostly for Internet-type applications, which probably has some sort of wireless".
As always in the technology industry, there are few precise definitions and lots of grey areas and exceptions. Different companies and observers pick & choose their own preferred terms to fit with their own market positions.
I've tried to use a consistent set of definitions in my work on mobile broadband and emerging devices. For reference, my thoughts (in brief) are:
Smartphone: device with an open operating system, full telephony stack, decent browser, and a size that means it can be held up to the ear for voice calls, without looking like an idiot. Typically with a screen size up to about 3 or 3.5 inches in width, although resolution may vary. Examples: Apple iPhone, E- and N-series Nokias, BlackBerries, most Windows Mobile devices
Obviously, they all have WWAN capability - predominantly 3G, as WiMAX isn't really mobile voice-optimised yet. Most now have WiFi as well.
MID (Mobile Internet Device): Handheld device with 4-8" screen, probably with resolution of VGA (640x480) or above. Various OS options, but not with a full PC-type specification and user experience. You'd be unlikely to do large spreadsheets on it, for example. Primary applications are generic Internet/computing tasks like web browsing, email, web-based productivity apps, social networking, access to some forms of content. Secondary applications vary, but perhaps navigation, voice, TV. Various form-factors (clamshell, slider, tablet etc), but generally too large to hold to the head as a "phone" unless you want to look like Dom Joly.
I make a distinction between "generic MIDs" - ie Internet-primary devices like Nokia N810 or the Aigo - and "application-primary" devices which also do mobile Internet stuff as a secondary capability, like the Archos 5, which I'd say is first and foremost a media player. Similarly, the Sony PSP has long had WiFi and is usable for web and VoIP, but is clearly a gaming product primarily. I reckon the iPod Touch is slightly too small to really be a full-spec MID but it's very very close to my (admittedly arbitrary but consistent) definition size-wise.
I'm expecting virtually all generic MIDs to have embedded WWAN - like phones, they'd be almost useless without it, and not necessarily support separate external modems with ease. Some will be 3G, some WiMAX and in future, some with both - and WiFi as well.
UMPCs (Ultra Mobile PC): Although the term is still used a lot, I'd define a UMPC as, essentially, a full-spec Windows Vista PC in a smaller physical form-factor, perhaps with a touchscreen and no conventional keyboard. Generally, UMPCs are quite expensive and may be tailored for particular vertical markets. Not, for the most part, massmarket consumer products. An example is the Samsung Q1. Almost all have WWAN capability built in, mostly 3G. All have WiFi.
Netbooks: These are probably the most prominent "small computer" form at the moment, exemplified by the Asus eeePC and more recently a plethora of competitors like the Dell Mini 9, HP Mini 1000, Acer Aspire One and numerous others. Basically, these are small notebooks, with a laptop-type form factor. They have 7-10" screens, and 80-90% size keyboards, with a mix of Linux and Windows OS's (no Mac netbooks yet....). They are typically intended for "lightweight" online and offline PC applications (eg web-based services, office applications) although they are generally not up to "heavy" gaming or corporate apps as they have less-powerful processors then full-size PCs.
Typically priced sub-$500, an increasing number will have built-in 3G or WiMAX, but despite predictions this will be far from "default" for a long time, as it will be an option rather than pre-configured. All have WiFi.
Sub-notebooks: The classic "small notebook" form that's been around for years. Normally a premium-priced, quite high-spec device such as various models of Sony Viao or Toshiba Libretto.
Notebook: Classic laptop, with a screen size of 11" and above. Mostly $500 and above. All have WiFi, and a small but (slowly) growing number will have embedded 3G and/or WiMAX.
Maybe in a year's time I'll need to review these definitions, but as I said above, it's always going to be possible to find exceptions or corner-cases.
Published December 2008: Disruptive Analysis research report on Mobile Broadband Computing. For details see here.