I have two rectifier diodes, a 1N4007 and a 1N5391. The first one is rated at 1A while the second is rated at 1.5A. What does the current rating mean? Will I damage/overheat either if I connect a device (such as a wall wart/barrel connector) that has an output of 2A or more?
The rating is the continuous current limit of the Diode. If you use these with a 2 A or 6 A power supply makes no difference ....it's the amount of load current that flows that is the limitation.
For example if you have a 2 A power supply but are only using a load that draws 1 A, you may be fine with the 1N1007 (though it will get warm/hot).
If you used a 1N1007 in an automotive project (the battery is capable of 100's of Amps) the same applies ....as long as the load current for your project (flowing through the Diode) is less than the continuous current rating for the Diode, everything will be fine.
So the current rating of the supply is irrelevant, it's the current rating of your load that must be considered, and must not exceed the Diode rating.
The key is to read the datasheet carefully. The 1A or whatever rating is under some particular set of conditions that may be optimistic for your situation. In the case of the 1N4007 (what I assume you mean, since a 1N1007 is an old germanium diode) a "1A" rectifier, the rating is specified as follows:
So we are talking average rectified output current with the lead temperature maintained at 75°C 9.5mm from the case. Note that they effectively define this as the ambient temperature which is.. rather dubious. There is an obvious danger if you have long leads and are running close to the maximum current. On the plus side, the actual RMS current may be significantly higher than the average rectified output current.
A graph on the datasheet shows linear derating to zero at 150°C (again that's the lead temperature 9.5mm from the case).
If you are running at higher current (or even close to 1A) I suggest you use a higher rated part such as 1N5407 or a Schottky 3A diode which would run cooler (because less voltage drop) eg. 1N5821. The 1N5821 datasheet I linked is a bit more honest in that it refers to Tl (lead temperature) rather than ambient temperature, however it is specifically the cathode lead and only 1/32" (0.8mm) from the case! There is detailed thermal design material in that datasheet, and it's pretty straightforward, just simple arithmetic.
Of course if your adapter can supply 2A but you are sure your load can never draw more than (say) 0.5A then a 1A diode is fine.
Read the rest of the datasheet.
Vishay's datasheet for 1N5391, for example, has these two charts:
So a continuous current of 2 A is not allowed. Even 1.5 A is not allowed if your operating environment does not keep the lead temperature below 75 C.
Up to 50 A current surges may be allowed if the surge duration is short enough and the surges are infrequent enough.