What happens if my neighbour sets his wifi SSID the same as mine?
As far as a standard 802.11 (Wi-Fi) client is concerned, the same SSID means it's the same network. This is how you set up multi-AP wireless networks that allow roaming between APs; you set them all up to publish the same SSID, so the clients know those APs are all part of the same network and the clients can roam between them as needed.
Some client implementations try to be smart about the potential problem you've raised (neighbors using non-unique default SSIDs), but you shouldn't assume that all clients will handle it gracefully and securely.
Having WPA2-PSK enabled on your AP will make sure that your neighbor's clients can't join your AP, but it won't keep them from trying (and possibly causing authentication error events on your router's system log). By the way, WPA2-PSK does not require the AP or the client to reveal the password to the other devices; that would be exceedingly lame. Modern authentication schemes use mathematical tricks that allow each end to prove to the other that they know something, without revealing to an impostor or eavesdropper what it is they know.
Having WPA2-PSK on your own AP won't necessarily keep your clients from trying to join your neighbor's same-SSID network. Your clients may prompt you for a new password for the network if they accidentally try to join your neighbor's AP, and the join fails because of the bad password. If your neighbor turns off security on his network your clients may actually successfully join without prompting. That is to say, I've seen 802.11 clients that were susceptible to security downgrade attacks, where just because they knew a WPA2-PSK password for a given SSID, didn't mean they felt compelled to always require that network to use WPA2-PSK; if they saw the same SSID without security, they might just join it and not bother with security. This is, of course, insecure and buggy behavior, and I would hope that the Wi-Fi Alliance's Wi-Fi certification testing tests to make sure that devices that pass certification don't do this, but you never know.
This can cause problems. Because there are 2 networks with the same name, you will only see one wifi network. Depending on which of the two networks is closer at the moment of connecting, thats the network that your device is trying to connect to using the password. If it is closer to your neighbours network, it will simply fails to connect and likely asks you to type in your password.
Do note that if you make a connection and then move closer to your neighbours wifi, things will keep working until you actually get out of range of your own wifi, in which case it will try to switch and fail then, prompting you with an enter password dialog.
I would strongly suggest to use a unique SSID for your network. Bear in mind that others can read this name, so it should not be an indication of to where you live, so burgulars won't have a clue where people live that have computers.
Also, Dlink, linksys etc routers often do use their name, but always have a small additive, such as Linksys-15326 which is unique.
Each of the SSID have a unique BSSID - so your device knows the difference once you are associated. Your issues will only be in the authentication phase. If they share a DS (Distribution System) - they will usually be managed by the same entity - hence your credentials would be shared. If they are unrelated, then your device may mistakenly try to join the wrong one depending on signal strength and it could fail unless it is open. This is a common “honey pot” strategy.
If they are on different channels, if you associate with the right one, it will not matter - what’s in a name. An SSID with a different BSSID is not the same.