How to tension the chain on a fixed gear or single speed bicycle:
Hang the drive chain on the seat stay, put the rear wheel into the frame as far forward/backward (depending on frame type) as possible, and transfer the chain onto the sprocket. Once the rear wheel is in the frame and the chain is running on the cog wheels, set yourself behind the bike with your head on the center line.
Pull the rear wheel back with one hand watching to check that it is centered in the frame. Do one nut up finger tight, carefully, so as not to disturb the rear wheel. Switch hands and do the other nut up with your fingers. Half tighten the nuts with the spanner. It is usually a 15 millimeter.
If the bike has a fixed gear, pick it up so the rear wheel comes off the ground and spin the pedals. If the bike has a freewheel you can do this by spinning the pedals backwards. The chain needs to be as tight as possible without creating friction that will waste energy and wear the chain and cogs prematurely.
To adjust the rear wheel's position, loosen one side and push the rim between the chainstays to one side to move the loose end of the axle a short distance back or forward. Retighten the nut. Loosen the other side and ease the rim in the other direction so the rear wheel is once again straight in the frame.
Test again and repeat the process, moving the rear wheel a short distance forward or back until the tension is in the target zone. You need to test by spinning pedals because neither cogwheel will be a perfect circle so the chain tension will vary as the cranks spin.
Once the chain tension is acceptable, tighten both nuts fully. If the bike is to be used for travel, lock the nuts with the spanner you carry for puncture repairs. If you use a long spanner in the workshop, the nuts may be harder to undo with your carry along spanner.