I've posted this information a couple of times, and someone suggested I just make it into a thread.
This will be based on hosting images from Photobucket, although the process is similar with other sources - you just can't add a picture that is on your hard drive or in a protected photo album (other than Photobucket), or no one will see it but you!
I don't have any affiliation with Photobucket and there are several free sites online to host images.
I have used Photobucket for years, both in forums, eBay and with family-
- Photobucket is free and only has annoying ads
- You can create "Albums" to better organize your images/videos and find them later
- You can password protect your albums, but images/videos posted in threads will still be visible to others
- You can create a password for an "Album" to allow other acces to just that album
- You can upload videos and share with family and friends - most e-mail accounts can't handle a 65mb video
- You can perform basic photo editing/re-sizing with Photobucket's tools
- You can add captions or tags to the images
- I've never had it "go down" and I've never used all of my bandwidth
*Note*
Portions of the instructions below would apply to posting images from other providers as well
To set up a free account, go to www.photobucket.com
Click the green "Join Now" button
Second page where you reveal all of your personal information 
I can't remember, but they'll probably send you an e-mail for confirmation - if so, just follow the instruction included.
When you first login, you'll have a white page (I changed mine to my favorite color) and the option to create an album - I would suggest it or it'll get very unorganized over time!
Once you're ready yo upload, look towards the section near the top
You have the following options:
"My computer" will ask for a file/folder location on your hard drive
"Web url" will take the file from a website (make sure you have or don't need permission)
"Mobile phone" will give you a number to text the image to and will put it in your album
"Email" will give you an e-mail address to send the pic to, similar to mobile phone
"Ask Toolbar" allows you to grab images and insert them directly, don't use it
You have the option for Photobucket to re-size the images as it uploads them
I'll often use the "bulk uploader" if I'm sending more than a few files
It gives you the option to re-size as well
Once the photo is uploaded and you click on it, you'll see this page:
You're looking for the code off to the lower left:
Email and IM - for sending a link via e-mail
Direct link for - inserting pic into post, i.e. GIRS
HTML Code - for web design or using the HTML option in GIRS post
IMG Code - for inserting image into other types of forums, i.e. Reef Central
To insert into GIRS, select the picture icon (similar for other online hosts)
Click "yes" to allow scripted prompts
Then paste the copied code from "direct link" into the pop-up box
There are other tricks but this should get you started!
HTH!
-Eric