HDMI is by far the most common connector for digital video signal transfer. As every stadard out there also HDMI was modified when requirements for video transfer increased. At the moment the most common versions are HDMI 1.4, 2.0 and 2.1. The newest 2.1 version is needed when you want to transfer 8K signal, otherwise it is an overkill. For full HD devices with resolution 1080p (1920×1080 px) the HDMI 1.4 works just fine.
If the HDMI adapter is active, that means if it converts the digital signal to analogue signal or vice versa, you need to pay attantion to parameters of video adapter in question. It is not automatically true that all the adapters will support full HD in sufficiently high frequencies 60 Hz or more. Such active adapters will need active power supply, which is usually 5V, so a common USB charger can be used. Other adapters are passive and they don't need an active power supply, for this reason such adapters are also cheaper.