What component did I remove from my old APC UPS? It's big and heavy. I have a link to pics in the details.

It's a transformer, but not all transformers are step-up or step-down, as the first answer suggests. Some are used for electrical isolation -- they are 1-to-1 (120VAC in -> 120 VAC out). This is useful when a fast voltage spike or sag comes through the AC power lines. The fast spike or sag will be essentially ignored by the transformer input and will have very little of the disturbance transferred to the transformer output, because it's maximized for efficient power transfer at 60 Hz (a very slow frequency). Because it's from a UPS, and they want to reduce the power line problems, it could very well be an isolation transformer (so be careful when measuring the output for the first time).

Transformer. It is use to step up or down voltages. It can convert 110 volt to 12 volt or 12 volts to 110 for example.

looks to be apart of the surge protection, or maybe it's the battery from the backup power source.