Usually Vim intellisense pops by itself, when it is most appropriate. You don't have to do anything in particular. But however, you can bring the dialog when you want, and hopefully it shows what you need at that time.
By default, intellisense pops up automatically. For instance in Java when you type . or in HTML when you type < it pops the intellisense dialog, with list containing the method names or HTML tags respectively. When you want to bring the dialog forcefully, press Ctrl-Space. While in Java it pops the list contains classes, interfaces, packages and member variables. In HTML it brings list of tags.
Help
If you configured the help, then you can bring the help. By default, the help for the selected item in the list appears in a window next to the intellisense window in two seconds. If you want to bring the help before the time, press Right arrow, this brings the help.
By default the intellisense brings the help window in two seconds. You can control the timing of the help by using this function:
:let b:dochelpdelay=3000
The value is given in milliseconds. This value is buffer specific. If you don't want the window to come automatically, then specify like this:
:let b:dochelpdelay=-1
But remember, if you press Right arrow, help window still appears.
You can change the size of the help window, by doing like this
let b:helpwindowsize = 120x240
Foreground and Background Colors
Intellisense pops the window in default colors. The help window appears in SysInfoBack as background and SysInfoText as foreground. You can change the color, by doing like this:
:call SetDlgColor("helpfg","blue")
:call SetDlgColor("helpbg","lightgray")
The first argument is the name of the window. It can take,
helpfg , help foreground window
helpbg, help background window
listfg, list foreground window
listbg, list background window
The second argument takes the color name. The color name can be from the RGB.txt which is installed along with Vim.
Keyboard Shortcuts
Ctrl-Space : Brings the contest sensitive list.
Right arrow : Bring help.
Ctrl-Right : Bring the next tooltip. (while tooltip is present)
Ctrl-Left : Bring the previous tooltip. (while tooltip is present)