If you're looking for this, then you are probably a member of one of two groups. The first group is just tired of having massive chat logs taking up space inside Skype, and wishes to clean things up a bit and remove pointless text from their program's history. This is fairly normal. While Skype will naturally hide old information, there's no real point to keeping months worth of conversations on hand. Unless you really like to reference old exchanges, just get rid of them from time to time.
The second group is presumably the more urgent one though. They're the group that has said something they didn't mean to say in Skype, and who now wishes to wipe it out. Well, there's a very limited option for that. If you're lucky, you can erase what you said with no ill consequences.
Deleting a Single Message
Since I assume the group that wishes to remove an ill advised remark is the most urgent, I'll cover them first. Skype offers editing and deletion options for a short time after a message is sent. The window isn't unlimited. You usually have about an hour to alter the message. If you're still within this time frame, then you're in luck. Of course, this won't help much in a live conversation. Despite recent advances in communications technology, it is still impossible to remove a memory from someone's mind. This will only work if they haven't read or received it.
If you wish, you can just right click the offending post and then choose the remove option. Confirm that you want to delete it, and it will be removed from your chat window and removed on your partner's window. Note that doing this will leave a note in its place, stating that a message was deleted. This isn't good if you're hoping to be incognito, but shouldn't really be a problem.
A smarter option may be to choose to edit the message instead. You can also do this through a simple right-click on the offending line. If they haven't received or read it yet, just change it into a different unrelated message and confirm your edit. There will be a pencil next to the post to note the edit, but editing a post isn't that unusual. They'd just assume you mistyped something.
Deleting Your Chat History
As noted, some of you may just want to clear up your chat history. It's never a good idea to leave things sitting around on easily breached or accessed online accounts. If you don't want people who can access your computer to read your chats, or you just don't want the information sticking around bugging you, then deletion is a good idea.
If you're just looking to clean things up, for organization or privacy related reasons, then you'll want to go to Tools -> Options -> Privacy and then click on the button to show the advanced options. There will be a dropdown bar that will let you pick how long it will store your history and a button next to that which can be used to clear your history.
Unfortunately, if the message was already read or you're past the point when it can be edited, then you're out of luck. The message can't be deleted. You're not done yet though, depending on just what the mistake was.
If you merely sent someone an inappropriate or insulting text (that they weren't supposed to see), then you can always be a little clever. Depending on your relationship with the person, you can probably just use a simple excuse, which may even be the truth. Just state that you sent the message to the wrong person, and that it would all make sense in context. This is probably the least painful method, if it will work.
If you sent something much worse, then you can always move on to lies. Thankfully, it's considered a fun prank now to jump onto someone's online accounts and change their information. If you say that your roommate/brother/sister/friend/enemy sent the message while you weren't watching, then it should work too. Everyone knows someone that would actually do it, so it's not a bad excuse. Hackers are also a current bogeyman, so you may be able to blame them for embarrassing messages, if the other person isn't too tech literate.
If worse comes to worse, you can always just hide your shame and block the contact. Deleting a contact on Skype isn't all that hard. You just need to right-click their name and choose to remove them from your list. If you wish, you can also go a step further and block them.
Source: Author's own experience with Skype
Image Credit: Wikimedia Commons/ZooFari