Have you moved away from your spawn point? The area around the spawn point can only be altered by administrator ("op") players. Either move away from spawn (about 20 or 30 blocks), give yourself op, or disable spawn-protection by adding
spawn-protection=0 to your server.properties file.
If you have done any of these, then your connection to the server is bad and you're experiencing desynchronisation between the server and client. The problem then is either your server acting up (is it modded?), or with your network connection / the network code at the operating system level and not Minecraft. Make sure your JVM is up-to-date at the very least, as I've experienced multiple minutes of lag loading chunks and opening containers when running the latest Minecraft on an outdated JVM on Windows.
If it's your server acting up, try restarting it. If you're running a modded server, make sure you're using the most recent stable version to avoid bugs that could cause an unstable connection.