I bought an ASUS VS239H-P monitor without speakers. I connect it to a laptop (Win 7) by an HDMI cable but no sound. Is it possible to make the sound from the laptop while it is connecting an external monitor or the additional external speakers are necessary?

Does the laptop not have internal speakers of it's own? If it does, and the sound is not on mute you should check and make sure there is a default sound set for when new hardware is connected. Go to control panel -> sound -> select the sounds tab. From there, there should be a list of actions that will play a sound when the action is met. Look for Device Connect in the list and make sure there is a default sound set for it. You can hit the test button to make sure it works.

The laptop has its own speakers. But, when it connects the external monitor by the HDMI cable, the audio goes to the monitor. The monitor, however, has no speaker but earphone port. Thus, there is no sound unless I use the earphone. If I disconnect the external monitor, the laptop makes the sound again. So, actually I want both of the laptop and external monitor have the sound when the laptop connects the external monitor. Does anyone know how to do it?

Is the monitors earphone monaural or stereo? If its stereo you can purchase an inexpensive set of 2.0 or 2.1 speakers to plug into the monitor. Typically an earphone plug is monaural (one ear only) and a headphone plug is stereo. If you plug headphones into the jack and get stereo sound, there you go. If it actually happens to be monaural plugging stereo speakers into it would mean only one speaker would have sound. Since monaural earphones are rare these days I would suspect its stereo. Your monitor manual will tell you in the specifications. Also if you use a DVI cord to connect the monitor (if that's possible) the laptops speakers should work.