Capacitive and resistive touchscreen technologies
Today's multi-touch screens are capacitive, unlike the resistive touchscreens from a decade ago. They work a bit like the way you can feel a static charge with the hair on your skin. The screen has a charge inside it, and when your finger comes near it, the capacitance of your finger attracts this charge. The screen can tell all the points where the charge is disturbed in this way, building up a picture of where you're touching it. That's why it can tell different fingers, and the more sensitive screens can tell fingers apart by their shape, and make a guess at how hard you're pressing.
The older resistive touchscreens just had two flexible conductive sheets with a gap in between them. When you apply pressure, it makes the sheets touch together, and the touchscreen hardware can find where that contact point is. Because of how it reads the contact point from the sheets, it can't understand more than one contact point, or a bigger area of contact.
This has two implications for a stylus. Styli for resistive screens need to be small for a precise touch, and can be any material (because they just need to apply pressure). A stylus for a capacitive touchscreen has to conduct electricity, but be covered with an insulator, to give it the capacitance (like your finger). They're commonly a conductive foam core covered with thin rubber. Usually the part you hold also needs to be conductive, to connect it to the greater capacitance of your body, but there are ways around this.
In addition, the tip needs to be wide enough to generate enough capacitance for the screen to register. In addition to the requirements of the hardware, most touchscreens also have some filtering in software, to avoid getting confused by sensor noise, tiny droplets of water, and other things.
For these two reasons, you can only use a stylus designed for capacitive touchscreens.
Foil hat time
You can make your own by sticking a small ball of aluminium foil to the end of any pen-like object with insulating tape. You just have to make sure your hand is in contact with the foil for it to work. This way, you can experiment with different sizes, to get the narrowest tip that works on your particular device.
If you still want something more precise, you'll need a phone or tablet that's designed to use a stylus. These phones have extra sensors in addition to the capacitive touchscreen, and they come with a special stylus that works with that sensor. Usually they detect induction fields (like a Wacom drawing tablet), and the stylus has an inductor built into it, but the details are different for different devices. Samsung's Note range of phones and tablets works this way, and Lenovo has a tablet (marketed at digital artists) that has a pressure-sensitive stylus like this as well as "palm rejection" software to avoid reacting to the bottom of your hand when you're drawing.