First, you need to check your connections. If the cable has a black box along it, you have a transformer to step the voltage down, before it reaches the console. If not, you only have to worry about the connection to the console.
Both should have a sticker or engraving to display the electrical requirements for input and output. Provided the plugs match, you generally have to match three things; voltage, polarity, and amperage.
Voltage
Voltage is the most important, as you need to match this exactly, on both ends. Not enough, and your system probably won't start. Too much, and it will probably fry itself.
First, you need to match the power output, from the wall.
Most countries use 240 volts. A few select places, such as the U.S. and Japan, 110 volts. You might even find the console/transformer supports both. If there is a difference, you will have to buy a step down/step up transformer. If you're not sure where to get one of these, you can try an electrical wholesaler, or any store that sells fuses.
If the cable used a transformer, you will have to match the voltage it outputs with the voltage the console accepts.
According to a couple of forums I visited, most notably, a post where a user is attempting to confirm an alternate transformer, the SNES takes a 9V power adapter. You should still confirm with your console. I can not stress this enough, voltage is easily the part I see most, where users mismatch the requirements and do irreversible damage.
If you're not confident in checking the numbers, again, take it to an electrical wholesaler. While they may not, necessarily, have stock of the required parts; they will be able to confirm if the adapter is correct, and offer support, if it is not.
Polarity
I've rarely seen this to be an issue, but if you use positive polarity when you need negative polarity, it certainly will be.
This should not be an issue, at all, if you are not using a transformer. In fact, I have only seen this be an issue with "multi-transformers", where you could actually change the polarity. When I have seen this mismatched, it led to frying the device and/or starting a large fire.
If you are using a multi adapter, included instructions should provide details on changing the polarity. Keep in mind that polarity is often represented by a symbol, as opposed to the words "positive" and "negative".
Amperage
Lastly, you will need to match the amps. Generally, this will be displayed as amps (A); but may be displayed as milliamps (mA). There are 1000 milliamps in an amp.
Generally you are allowed to exceed this value, but some argue that you should not go too much over the requirement.
Best Answer
Generally speaking, using generic chargers is a bad idea. They're not as reliable as the official ones. With that said, theory says you should be okay using it, though it's not ideal for the DS.
The output voltage of an official charger it's 5.2V and the output current 450mA. Your charger is 5V and 1000mA. Lower voltage means the DS will not be charged as effectively as usual, but it shouldn't cause any damage. Higher current means the DS will have enough current to pull from, with the adapter having some excess, and there shouldn't be any problems either. So in theory the only issue here is that you're not charging your DS as effective as possible.
The problems usually come when using more voltage than required or lower current than required. Here is a more detailed explanation of the whole chargers issue.