Smoke is caused by firewood not burning completely and is due to either a fire that is not hot enough or excess moisture in the wood itself.
When you first start a fire with kindling you can expect some smoke as the fire temperature is initially relatively low. As the fire takes hold, the temperature of the fire rises and smoke will reduce. Once the fire is going well there should be much less smoke and at this point it is important to keep fueling the fire with wood to keep the temperature hot enough to continue to stop smoke. Letting the fire die down and then adding wood will increase the chances of smoke as the fire has to get hot again. When adding wood to the fire place it directly in the middle of the fire on the top, this will ensure the newly added word burns more efficiently.
If you are still getting smoke it is almost certainly due to excess moisture in the wood itself. Excess moisture may be due to two factors:
Wood that is not properly dry before burning will generate a lot of smoke. This may be because it was left out in the rain or poorly stored in a damp location. Even wood that you buy from shops that is supposedly ready for burning may contain too much moisture for an efficient smoke free burn.
Unseasoned wood (wood that is too "green") should not be burned. You cannot just cut down a tree (unless it has been dead for a long time) and expect to be able to burn it without smoke. Green wood contains sap which is mostly water. Water in the wood has to be evaporated before the wood can burn and so massively impacts the burning temperature of your fire and this leads to smoke.
You can buy a moisture meter online for about £10. These are very useful devices and can help you to select wood that is ready for burning. We've used them to check wood sold in shops and the results are very surprising. Many string bags of wood we tested showed that the moisture content of the wood was too high for a smoke free burn. Sealed bags of kiln dried wood faired much better in our tests and we'd always recommend these bundles for fire pits. They are slightly more expensive, but well worth the extra money if you want a smoke free fire.