How do I stop my neighbor from using my IP address?

Summary: If it's certain that a neighbor is using your IP address without permission, you need to take steps. But first, you need to make sure.

My neighbor used my IP address to send out emails. How does this work? Can I change my IP address? I have googled this topic and the answers are very confusing.

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It all depends on how you think he got your IP address, and in fact whether it's really yours at all.
There's at least one common scenario where this does represent a true security risk, and we need to nail that down. But there are also scenarios where it's not really an issue in your control.

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If your neighbor is truly using your IP address, that implies he's connected to your network. That's a bad thing, unless you trust him and have explicitly allowed him to do so. I'm guessing from your question that's not the case.

"If your neighbor is truly using your IP address, that implies he's connected to your network."

By far the most common reason for this is simply that you have an "open wireless hotspot":

you have a wireless router or access point the wireless connection is not password protected your neighbor is in range of your wireless, and is using it

It's important to note that your neighbor could be completely unaware - it can happen accidentally or if he's not paying attention to what he's connecting to. I would not assume malicious intent on his part, unless you have additional information that says otherwise.
The fix is very simple: password protect the wireless connection. Configure the wireless router or access point to use WPA which then requires a password to be able to connect to the wireless network. Of course don't tell your neighbor that password.
No one except those you've authorized will be able to connect to your network.
As a bonus, your connections will be encrypted, and no one will be able to "sniff" them either, as is possible at most free wireless hotspots.

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There are a few other reasons that he might appear to have sent email "from" your IP address.
You could have misread the email headers - your IP address, as the destination when you download your email, may well appear within it, in addition to the IP and email address that the mail was sent from.
You and your neighbor could have the same ISP, and both have "DHCP" assigned IP addresses - which can change. In fact, what's your IP address today could easily be your neighbor's IP address tomorrow.
The email might not be from your neighbor at all. Your machine could be infected with a spam-sending virus or bot, and it could be "spoofing" the from address so as to appear that it's being sent by your neighbor.

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Can you change your IP address? Well, that too depends - this time on what kind of IP address you have.
If you've been assigned a static (unchanging) IP address, you'll need to contact your ISP and coordinate getting it changed with them.
If you have a dynamic IP address (DHCP, above), then you can try to change the address by simply unplugging your internet-connected modem and/or router, waiting a while, and plugging it back in again. This way it will ask for a new IP address. The problem here is that there's no way to force it to be different than whatever you had before - you may get the same IP address again. In fact, many ISPs explicitly try to give you the same IP address for "a while", even when you're not connected for a time, as it can be slightly more efficient for them.