Does this switch card keep data passing if the power is cut off?

Yes. It uses proprietary latching switching technology, so data can continue to pass through even if power is lost.

What type of connection does this Black Box Pro Switching Gang Switch card support?

This card is the DB25/RS-232 A/B switch card, designed for RS-232 connections.

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Black Box Pro Switching Gang Switch - 2U - RS-232/DB25 - A/B Card - Latching - SM265A

A/B Switch Cards are available for RS-232, video, fiber optic, and CAT5e/CAT6 connections. The card featured here is the DB25/RS-232. Use the system for both new and legacy applications. You can add cards for fast CAT6 10-GbE networking while keeping connections for older RS-232 and 75-ohm video connections.
For fail-safe operation, all the cards (except those noted) use proprietary latching or micro-mirror relay switching technology - which means data can continue to pass through even if the power has been cut off. Because the cards perform Layer 1 switching, the system operates independently of data rates, protocols, formats, or signal levels.
The non-latching fiber cards fall back to a Port C (chassis) to Port A connection when the power fails. When power is restored, the non-latching switch card reconnects Port C to the originally selected port (A or B). These types of cards are typical when you want a fallback or failover connection during power outages.
Each chassis also has a keyed switch so you can enable/disable manual switching from the front-panel toggle switches.