You can run safari in Xcode's simulator and it should accurately emulate iPads and iPhones. Another thing on the market that I've heard good reviews for is Ripple for chrome.

The iPhone/iPad simulator that comes with Xcode includes Safari. If you run Safari in the simulator, you can view your website and it should appear the same as it would on a real device. This may work for general layout testing. But since it is a simulator, it is possible that not every single bit of functionality will be exactly the same as using a real iOS device.

If you are writing a website and you need to verify that it looks proper on a given device, then you need to test your website on that actual device. Testing with real hardware is part of the price of doing business.

And yes, you need a Mac to run Xcode.

XCode does come with a simulator for the iPad and iPhone.

You can also use Safari on OS X to debug websites on your iOS device.

There's no good substitute to testing on an actual device.

Real devices have higher display densities, meaning that pixels are smaller. If you don't test on a real device, you may not realise that your design includes text that is too small to read or buttons that are too small to tap.

You use real devices with your fingers, not a mouse. This means that the accuracy of your taps is much lower and what you are tapping is obscured by your finger. If you don't test on a real device, you may not realise you've introduced usability problems into your design.

I have been using Mobilizer, which is an awesome free app

Currently it has default simulation for Iphone4, Iphone5, Samsung Galaxt S3, Nokia Lumia, Palm Pre, Blackberry Storm and HTC Evo. Simple straightforward and effective.

You can use the Ripple emulator on Chrome.

Both Chrome and Firefox now have built-in emulators. They aren't perfect but are good enough that can get you almost all of the way before testing on an actual device. The best part is if you like the browser's developer tools (Chrome, Firefox), you can use them while emulating.

To get the emulator: [Ctrl+Shift+M] and select the device that you want to emulate. You might have to refresh the page, esp if you have anything that depends on script that executes on page load.

Internet Explorer also has a device emulation mode. F12, then CTRL+8. It's not quite as straight forward as the Chrome Mobile Device emulation, but does allow you to simulate geolocation:

EDIT 2020: Most of these are basically just to test resolution stuff, some of them even outdated, sadly, mobile browser development went sideways with desktop (especially in Apple), therefore one can't really "emulate" a real phone with these as mentioned with comment.

To emulate real phones, often the best choice is to download a desktop app which, for Windows, is usually paid/freemium, on Mac just use the Xcode one (but I doubt Mac people are looking for this Q/A).

Freemium online easy to use that I found recently is Appetize.io it seems to really render the screen according to network, but honestly I didn't really dig into whether it also has identical features and indentically missing features as real iOS.

Word of advice:
before release, always test on the real device :)

I've encountered cases when even the iOSes themselves behaved differently on 2 iPhones...

Online simulators / emulators I use

1) recombu

Fine simulator which - unlike resizing browser window to mobile phone dimensions - acts same as a smart phone. Don't be confused that you can't edit address bar in safari - just open deveolper tools (usually F12) and rewrite iframe's source URL to yours.
Link: http://recombu.com/mobile/interactive/ios7-demo/

2) responsimulator

Seems to work like recombu, but you can open url directly by text input and you can zoom in/out.
Link: http://www.responsimulator.com/

3) transmog

This one seems to process the webpage, but it emulates old iPhone - still handy sometimes.
Link: http://transmog.net/iphone-simulator/mobile-web-browser-emulator.php

X) google it / internet search for it

Always use google (or other internet searchers) to check for other simulators/emulators and new versions.
Link with example google search for this one:
https://www.google.cz/search?q=online+iphone+emulator

Browser device mode

If you open your browser's developer's tools (in Chrome F12), there will probably be an option to toggle device mode (in Chrome it is the little smartphone icon at top-left).

After choosing this option GUI will change and will provide option to select device to simulate (in Chrome it is at the top - select option "Device"), after selecting device, refreshing the page is often adviced to ensure simulator's accuracy.

For now i think best emulator is https://app.crossbrowsertesting.com

It has real sizes and virtual keyboard (that is the most important thing) , zooming events...

Also https://appetize.io/demo has same things but it has time limit.

I use mobile-browser-emulator chrome plug-in which is has iphone device types. It actually uses user-agent and size of device on which based responsive pages are rendered