HDCP (High-bandwidth Digital Content Protection) is a system designed to prevent the unauthorized copying of digital video content. It involves checks on all key devices and components in the chain that allows you to view the content, including the monitor. You can check yourself whether your monitor is highly likely to be HDCP compliant, but you may need to check with the manufacturer to be certain.
HDCP works only with digital connections and involves encrypting the data as it travels between devices. Each device must support HDCP and thus be able to encrypt or decrypt the data, depending on the device's purpose. The point of HDCP is to stop users from connecting a playback device to a recording device to copy content without permission.
When HDCP was first introduced, few monitors supported it and it was a safe bet that unless you specifically knew a screen supported HD, if almost certainly did not. Today the reverse is true: the standard has caught on to the point that if a recently-manufactured monitor meets the general requirements for high definition, it likely supports HDCP as well. If your monitor does not support a digital connection (either HDMI or DVI) and a resolution of at least 1920 x 1080 pixels, it will certainly not be HDCP compliant. If it does have these connections and resolutions, the chances are it is HDCP compliant.
Other than trying a monitor out -- in a computer setup where you know all other relevant components are HDCP compliant and the source material is HDCP protected -- can you confirm for yourself that the monitor is HDCP compliant. DCP, the organization that oversees HDCP licensing, lists some compliant equipment (see Resources) but the list is not exhaustive and concentrates mainly on devices other than monitors. Check the instruction manual if you have it, or the product listing on the manufacturer's website, to see if there's any mention of HDCP compliance. If you are still uncertain, you will need to speak to the retailer or contact the manufacturer directly to confirm.
Remember that having an HDCP compliant monitor isn't sufficient in itself. You'll also need all other relevant components to be compliant. With a computer this usually means both the player (that is, the Blu-ray disk drive) and the graphics card. HDCP compliance isn't an issue with the cable itself as it simply carries the digital data, which will already be encrypted. However, you must be using either an HDMI or DVI cable, as the encryption doesn't work with analog signals.