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I want to put my router in a box for security reasons (it's in a common area of the building) and I was wondering in what ways would the box's material affect the signal's strength. It'll be either metal or wood.

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If enclose it in a metal box, you could end up making a Faraday cage, which would cancel out much of the signal.

Wood shouldn't be a problem (2.4 GHz can go through walls), but it will still impact the signal strength (though probably not as much as the walls currently separating you). Make sure you have enough power to get from where the router is to where you are, and you should be fine.

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Metal will interfere with wireless signal, wood will not (noticeably) interfere. Make sure it is vented for airflow.

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You can mount a router inside of anything, as long the antenna are still exposed. That said, find a decent router (like a high-end Netgear), that uses Coaxial connectors for the antennas. Then you can connect the external antennas outside the box anywhere you wish.

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The more material you put between your devices and the router the weaker the signal can become.

Here is an article that talks about various materials and their impact when put between the device and the endpoint.

Material summaries

Clear glass -4db

Sheetrock and insulation about -2db for each intervening wall.

Plywood -5 to -12db

Brick -28db

Metal -32 to -50db

Decibels are a measure of signal strength and loss. If you're not familiar this will be helpful

I would look for other ways to secure the device than a box. If you must use an enclosure, the above list should help you choose you materials appropriate to your purposes.

To understand antenna's better here is a good article about antenna theory with an emphasis on Wifi

Hogstrom
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I got a moment to think... and what really matters is the antennae not being enclosed in the box.

If the antennae aren't enclosed and the only thing inside the box (which would have holes for the antennae) was the router, then it doesn't matter whether the box is made of metal or wood, as long as it was properly ventilated.

Even if the metal box had a Faraday Cage effect, since the antennae are outside it shouldn't interfere.