No problems at all. The two countries are friends especially due to the pipeline bringing oil from Azerbaijan to Turkey via Georgia. Nationals of either country can visit the other without visas.
There are two active border crossings and I believe one inactive one.
The main one is on the Black Sea coast and is very busy and has been modernized on both sides of the border. The Turkish side is Sarp (Hopa district) and the Georgian side is სარფი (Sarpi). Many trucks cross at this border between Europe and Georgia and beyond to Armenia, Azerbaijan, Iran, and even some Arab countries (well at least before the Arab Spring). The highway passes through here from Samsun to Trabzon on the Turkish side to Batumi on the Georgian side.
There is also a very small rural one. The Turkish side is Türkgözü (Posof district) and the Georgian side is ვალე (Vale). The highway here connects Kars on the Turkish side to Akhaltsikhe on the Georgian side. This border crossing is very quiet on both sides. Edit: scheuer's answer has opening hour details.
At least for tourists and I'm pretty sure for truck drivers too there is no corruption on either side. You can cross at both points on foot with no problem. The Georgian customs and immigration workers at Sarpi are among the friendliest I've ever dealt with. It usually takes a while to cross here because many busfuls of passengers cross and due to Armenia's borders with both Turkey and Azerbaijan being closed, everybody has to pass through here.
The rural crossing is quite quick to cross. The road has very little traffic which will affect you if you're hitchhiking like I was. There are very few facilities in the villages on either side but the police in Vale on the Georgian side are extremely friendly and helpful. There are many stray dogs around the rural border which worried me because I was on foot and there was nobody to give me a ride, and there were no businesses or houses close to either side in case I were to need help. But even the big ones turned out to be very docile so no problem.
On Google Maps you might be able to make out another rural crossing at Kartsakhi Lake / Lake Aktaş. This one is not operational and I do not know whether it has ever been in use.
There are international bus routes passing through the Black Sea crossing connecting at least Batumi and Tbilisi to Trabzon, Samsun, Ankara, and Istanbul. There are even buses connecting Tbilisi to Thessaloniki and Athens in Greece! There are no scheduled buses crossing at the rural crossing. There are no rail crossings between the two countries but Georgia does have rail crossings with all of its other neighbours. (The one to Russia is not operating as of May 2012 however).
Turkish international buses were quite nice and quite cheap. Individual TVs on the back of each seat on at least Luks Karadeniz. I took Samsun to Tbilisi and was told it would take 10 hours but instead it took 15 hours: 10 from Samsun to the border and 5 from the border to Tbilisi (roughly).
If you cross on foot most people take a very cheap taxi from the border to Batumi. I don't know details for the Turkish side. Both countries are famous for their hospitality though. As for the rural crossing, it's a fun challenge to cross on foot as there are no connections and few trucks and cars. Nevertheless if you don't mind waiting in quite beautiful settings it is very easy to hitchhike on both sides though getting to Kars might take a while if you're not young and blonde (-: