Okay, a quick run-down of the items you mentioned (just my take, others will add their thoughts I'm sure):
Thermometer - shows the temperature of the water. Most fish have a range of temperature they're healthier at, this will show where your water sits. Not essential but highly recommended.
Heater - keeps the water at an appropriate temperature all year round. For tropical fish, I would say a heater is essential.
Freshwater substrate - goes in the bottom of your tank, in the form of usually sand or gravel. Different fish (for example corydoras) prefer sand. If you're going for a heavily planted tank some use soil in the substrate. It gives the fish a more natural environment, and if you have plants it gives the roots somewhere to grow. I would say substrate is essential.
Air pump and air stone - the pump puts air into the stone and gives you bubbles in the water. Not essential, some (such as me) like how it looks.
Hydrometer - I would not say this is essential.
But the very most essential item in your tank is a filter. This is where the essential bacteria grows to take care of the waste the fish produce. It also takes some of the particle waste from the tank to keep your water clear. Without a filter your water will quite quickly turn toxic to fish, they may look all right for a short while, you may be able to do water changes to keep your fish alive a bit longer, but without a filter I'm afraid you'd be looking at fish deaths on a regular basis.
Must go now, but I'd be happy to contribute further when I have time. In the meantime, please read the Cycling a Tank link in the ribbon at the top of this page, it will introduce you to the nitrogen cycle and explain why filtration is essential.