'Full HD' means 1080p resolution (1920 * 1080 ) which is basically the minimum/standard for desktop monitors nowadays. For gaming it's a good resolution choice.
Not sure if that monitor is particularly good or not, hopefully someone else can advise!
For gaming the main considerations are: size, resolution, frequency, response time, input lag.
Size of course depends on the space available for it to sit in and personal preference, 24" is a good option unless you know otherwise.
Resolution we already covered - you're 1920*1080 which is fine, many here game on higher resolutions but it's not really for the entry-level player as they cost more and need better computers to run.
Frequency is how often it can draw a picture - 60Hz is 'standard' but some panels go higher which can give a smoother playing experience. Again, I'd not worry about this for your needs, this is likely 60Hz which is fine.
Response time is how quickly it can change what it's pixels are displaying - too slow and you get a 'smeary' image in fast-moving scenes (think long-exposure night pictures). 5ms gtg suggests this is acceptable but nothing special here - though witthout a proper review it's hard to tell how good or bad it is I'd not worry too much.
Input lag is how long after your computer sends an image it starts drawing it - how long it spends thinking about it. I have no idea how long this monitor will take here, though it appears to use a TN panel which normally are not too bad in this regard.
Finally panel - where most people start. TN is the cheapest option, but often also the best option for gamers as it often supports higher frequencies, has decent response times and low input lag. However it has the worst viewing angles (if you go off to the side, above or below the picture looks really funny), not great colour reproduction and it doesn't have great contrast (max difference between black and white, thus how different all shades are in between)
Alternatives are IPS (think apple screens) - great viewing angles, decent colour reproduction, normally fairly low input lag. However, contrast ratios are bad, prices are higher, frequencies are almost always 60Hz and response times are slower.
Finally VA. This is the king of contrast ratios coupled with decent colour reproduction. Normally it isn't great on frequency or response times (some exceptions, look for the 240Hz Eizo in this forum, solves both problems)
For a first high definition widescreen gaming monitor I'd certainly pick a TN panel at 1080p... which one I don't know!