Android - Is it possible to root my phone without installing a custom ROM?

Short answer

That very much depends on your device, as the rooting process often is device specific. Following up to the rooting tag wiki will give you some basic info, and also provide a link to our rooting index, which might even already include a link for your device.

TL;DR

Basically, rooting just means to push a binary (su) and a helper app (SuperUser/SuperUser or SuperSU) to the system partition. So as Compro01 wrote in his comment on your question, you're "customizing your ROM" somehow.

On most devices that's all you need, though how that's accomplished is a different thing, as it's a little chicken-and-egg: You want to gain root permissions on your device – and for that, you first have to push those two files to /system, which only can be done with root permissions. So in most cases, this requires some "hacking", which is what Sparx refers to in his answer might cause your device to be unusable. I rooted several devices already, and when sticking to the instructions the device should not become "unusable" (though I admit there might be rare cases). I had it once a device became instable (apps crashing, device rebooting itself multiple times a day), but that was easily cured by a factory reset.

In theory - yes.

In practice, it varies from device to device - a very common way of an end-user obtaining root is to flash an insecure kernel. This can be a standalone kernel or a pre-rooted kernel + ROM combination. For certain devices, there are also available a number of 3rd party "one-click rooting" apps.

That being said, rooting might cause your device to be unusable (if not done correctly or something unexpected occurs), so read the instructions carefully before proceeding and ALWAYS backup your data to an external source like Dropbox or to your PC before you begin.

Here's a link (unverified) on how to root your Verizon Stellar.

The answer is YES .

It is not mandatory for you to install custom ROM for rooting your device. ROOTING process is a actually YOU ARE GETTING ALL THE PRIVILEGES of your phone . So it is not necessary to install Custom ROM

Steps to Do Rooting :

1. Find root files for your Android Device on the internet(Every phone has a different root file ). Place it in your SDCARD
2. Reboot your phone in the boot loader mode.
3. Select file from the boot menu
4. Hit install.

I can help you in your rooting process if you tell me which device you are using.

here is an Unverified way to do it (Samsung Steller).