Don't lie. You know you've done something like that. Granted, you're probably more sane than that. Well, at least 24% of you. Here are a few methods I myself use to find those mystery characters. This guide is mostly Google Images-centric, since I only use Google images. Please keep in mind, you can right-click the small images to open them in a new tab.
Please Note: If you're too lazy to do any of this, and insist on relying on other people to tell you who it is, pressing Alt+ F4 will probably solve any problems.
Check the file name within the URL:
This is the easiest, and the first thing should do. More often than naught, you'll have clues that relate to who the character is. If you're lucky, you'll get the name of the character or the identity, and from there, you just need to use Google.
Since I know what a tsundere is, I simply look up shugochara, annnnnd.... OMG She's from an anime called Shugo Chara! I click the first link, and she is apparently Hinamori Amu! Yay! That was easy.
View the website where you found the image (For Google Images):
"Oh no! This image only has numbers for it's filename! What do I do?"
Simply go click the image, and a page similar to this appears. Click the "Website for this Image" link, and search for the image on that page for clues. This website lists a few names, so I just started searching for each one. Since the other characters didn't match, I'm assuming that they are from La Corda d'Oro.
_______________________________________Use a Reverse Image Search tool:
So, you tried the above, they didn't work, and now you're ready to go explode and blow up the interwebs or something. HOLD IT. There's one last thing you can try. Reverse Image Search tools are exactly as they sound, they search images in reverse finding you the places where the image showed up on the internet.
TinyEye: Reverse Image Search Tool and clicked the search button. It only yielded one result, and it was in French! I don't know French. However, I still found a good clue while I was scrolling down the page.
OH LOOK! WORDS. A good rule of thumb: if there are words on it, search them. Upon searching the words, I found she is Nagi, a character from an anime called Kannagi: Crazy Shrine Maidens.
Nota Bene; Google has a similar feature on the Google Images homepage! It yielded more results for that image than Tiny Eye, but I didn't find many clues until I went though a few links.
_________________________________________Anime Character Database:
Now, if you tried all of the above, and they didn't work, your last resort is the Anime Character Database.
I have had little luck using it, but something is better than nothing, no? The Anime Character Database allows you to search by the following attributes, respectfully:
Gender - Eye Color - Hair Color - Hair Length - Apparent Age - Animal Ears
Input the specifics in the top bar in the quick search, but remember that the colors and attributes might not be 100% accurate. When I searched using "Female - Purple - Blue - To Waist ". I did not find any pictures of Nagi. However, when I used only "Female - Purple - Blue" I was able to find her.