When you type some letters, you can activate this completion mode for LATEX commands and normal words with ->-> or the keyboard shortcut Ctrl-Space. Kile first reads the letters from the current cursor position to the left and stops at the first non-letter character or a backslash. If this pattern begins with a backslash, Kile will enter completion mode for TEX or LATEX commands. Otherwise it enters normal dictionary mode, where you will not find any LATEX commands. Depending on the chosen mode, a completion box will be opened. You will see all commands or words whose beginning matches the current pattern. You can navigate with the cursor keys through this list and select one entry with Enter or a double click with the mouse.
When you push the Backspace key, the last letter of your pattern will be deleted, and the completion list may grow. On the other hand, if you type another letter will expand the pattern and the visible word list may shrink.
If you decide not to select any of the suggestions, you can leave this dialog with Esc.
You will see that all commands are written with a short description of their parameters. These descriptions are of course stripped when you select a command. Optionally you can let Kile insert bullets at these places, so that you can easily jump to the these positions with ->-> and insert the parameter you want.
Go to ->->+Complete to configure one or more of these lists. You can choose different word lists for TEX and LATEX commands and dictionary mode for normal words.