A simulator is software appearing on a computer that acts like the target environment. A browser that is the same size and the same official standards as the target browser, but not the actual code the target device is running. An emulator is running the actual code, so from a technical perspective you can have confidence in the fidelity.
MobiForge has published a useful guide to actually getting over a dozen emulators to run. Many of these are buried under their developer sites, or have some other trick to get running, so the guide is a good thing:
Web-Based
dotMobi Online Emulator - focus on common phones; likely to be a simulator not emulator. Run by YoSpace.
Opera Mini Simulator - Opera's J2ME browser.
Bolt Browser Simulator - Intended to sell the product, but useful to check how things might work on a small-screen, scroll-and-select browser.
Device Anywhere - Actually a new class, a virtualization system. Actual hardware is disassembled, things are soldered to it, and it's strapped to cabinets. You get to press buttons and it goes over a real network, and gives you the screen output. Free trials available with some developer programs, otherwise a fee service. But useful.
Applications
Nokia Browser Simulator - generic Nokia phone and gateway
Openwave Phone Simulator - Openwave's simulator (probably actually an emulator)
BlackBerry Simulator - multiple different BlackBerry emulators
Yospace SmartPhone Emulator - displays multiple phone instances at the same time but may not have the best fidelity to actual devices
iPhone/Pre Emulator - Windows only desktop emulator, emulates both iPhone and Pre.
Palm Project Ares - Full web development environment for WebOS in the browser.
Prototyping / Wireframing
iPhone screen projector - "tethers" with your mac to display content from your desktop on your iPhone. Requires a free iPhone app.
iPhoneprototype - Firefox plugin for iPhone mockups.
Adobe Device Central - Part of the Adobe CS product line, lets you see what the design might look like on various mobile device screens. Includes non-whites, non-blacks, glare, etc.
Full Design Suites
MIDS - flow charts, navigation, wireframes, rendering, and even a physical test device