In the movies and TV shows they show that the only way for a cellphone to not be traced is to take the battery out (and optionally break it in two :-).
But if this is true, then that means that even in aeroplane mode one's cellphone is pinging the cell towers -- how else could a cellphone be traceable?
Corollary: is a cellphone traceable by just the SIM or the SIM and the IMEI number of the phone?
In flight mode -- at least in every mobile phone I've come across that has that feature -- all the radios are disabled[1], so the device is inhibited from transmitting any signal. If you power up an iOS device while already in flight mode, it doesn't even initialise the SIM card until you switch off the mode (if your SIM card has a PIN code set, it won't prompt for the code if powered on in flight mode). So while in flight mode, a mobile phone cannot communicate with any other device, and that includes base stations.
To answer the second part of your question: the phone is traceable by its IMEI, which (in theory at least[2]) is unique. Of course that doesn't mean you know who owns the phone. The SIM card number is also unique, and is a better target for tracing a user as it will be tied to an account with the provider[3].
[1] On some devices, you can enable flight mode and then selectively switch on some radios. On iOS, for example, one can use Wi-Fi and Bluetooth in flight mode if these are manually selected 'on'.
[2] In practice they are almost unique. I have seen devices leave a factory with duplicate IMEI numbers by mistake. It's also possible with some hardware to re-program the IMEI number.
[3] In some jurisdictions (e.g. the UK), it is possible to buy a SIM card from a shop and use it without the provider knowing your identity.
Spy movies and TV shoes make a great deal up.
They are in no way a credible way of telling what is really possible.
As "Airplane Mode" is a way to follow the FCC's ban on "inflight use of cell phones because of potential interference with ground networks"1, someone would have to tinker with a phone's software to disable correct "Airplane Mode".
But seriously, before you consider movies legitimate sources spook info, consider how many movies have x-ray satilites, and can somehow rotate the view a satellite to on a horizontal plain.
