I was recently asked by a friend if it's possible for someone to find someone else's location if they didn't have the location services on, on their phone. I told him I'm not sure but I didn't think so. The assumption is that the person your looking for has a phone and you can use that to find them, but if the location services are off does that make them invisible? Is it possible to find them?
Since nobody has bothered to write an actual answer: the phone company can certainly locate you. The precision isn't great, because it's based on how strongly your phone's signal is picked up by nearby cell towers (which will vary depending on things like objects in the way and is going to be a bit fuzzy anyhow because the signal strength measurement isn't designed for precise range measurement). If there are a lot of towers nearby, then you can probably get precision down to something like "that house" or possibly even "that wing of the house", which is probably good enough (it would generally work for police dispatch, certainly). If there's only one tower actually in range - which is uncommon in a city but happens out in the country sometimes - then all they can get is that you're somewhere in the tower's coverage area, and an approximate distance (presenting a rough ring around the tower).
It's worth noting that even when not expressly trying to triangulate you, the phone company will always know approximately where you are by which tower your phone actively communicates with. The system is designed to automatically and near-instantly hand off communication to another tower when you get closer to it, and this can sometimes be used to track people with loose precision. A common example is a phone in a car going down the highway; you can track about where the car is, and how fast it's going, by watching the phone switch from tower to tower as it moves. If there's only one highway along that route (and it's going too fast for any side streets that may parallel the highway), you can be pretty sure the phone is in a car on that highway, and also know roughly where it is (enough to set up a roadblock, not enough to identify the specific vehicle without further work).
Note that while turning off the cellular radio (Airplane Mode) will make it impossible to track you this way, taking out the SIM card will not. The phone presents two identifiers in normal use: the IMSI (subscriber information, which comes from the SIM card and ties the phone to your phone number, account, etc.) and IMEI (equipment information, which is stored in the phone's radios and identifies the phone hardware; it's possible to use it to track or block stolen phones, and enables emergency calls even when there's no SIM present).
In addition to the cell phone company (which will generally cooperate with law enforcement, certainly if they have a warrant), people (including law enforcement) with fake or portable cell phone "towers" could track you if you're close enough to the device for it to pick up your phone's signal. (The one used by law enforcement is known as a "Stingray", but you can build one by hacking the "femtocells" that some carriers will provide to improve signal in your house or whatever.) It's more common to use such things for intercepting phone conversations and data transmissions than for tracking, but at the very least the fact that one of them sees your phone can localize you to a specific area.
Yes, they can.
One major flaw in the privacy of mobile networks is a protocol called SS7: this is the mobile carriers' network. It was designed as a trusted network: once you're part of it, you're assumed to be a benevolent carrier. When you're part of the network you can ask for the base station ID where a mobile phone number is currently booked in (and much more). This gives you the approximate location of the phone.
It is not easy or cheap for a normal person to gain access to this network but there are companies participating in this network that sell phone locating services, abusing their SS7 access.
For more information about SS7 please watch the great talk SS7: Locate. Track. Manipulate. from the 31. Chaos Communication Congress.
Your mobile carrier may protect your privacy against other participants of SS7 but many carriers don't. Check the SS7 Map to know if your carrier protects your privacy.
A completely different way to locate you is if an app on your phone has access to the list of Wi-Fis currently seen by phone. There are databases with Wi-Fi SSIDs and their location – so if the app has access to such a DB and your Wi-Fi system, it can quite accurately locate you.