how to connect mac mini to HP 25er monitor

Your Mac has a DIRECT HDMI port. So as responders have said, connect directly from your Mac's HDMI port with a high spec HDMI cable.

But not just any cable.

HDMI cables you want for HDMI-only Displays (higher resolutions than 720p TV sets) are marked as Certified with an anti-counterfeiting tag and are labeled:

"PREMIUM High Speed HDMI cable" or that + "with Ethernet" (up to 4K at 30Hz) --OR--

“ULTRA High Speed HDMI cable" or that + "48G" (supports higher resolutions and backward-compatible)

Cables with No Certification tags are good for your standard 720p TV set, and not much more.

Cables 'shipped in the box' with many displays can be "lowest bidder" cables that are not up to high-end display use. Cables with no markings 'found' behind your TV set are unlikely to be good enough.

HDMI was invented for HD TV sets. it works great at its original resolution of 720i or 720p. At higher resolutions, it quickly develops issues that are complex to solve, and the cables and adapters required to solve are NOT intuitive.

Using the Mac mini HDMI port and an HDMI cable to connect to the HP 25er HDMI port seems easiest.

• https://h10032.www1.hp.com/ctg/Manual/c05105530.pdf
• Mac mini (Late 2014) - Technical Specifications - Apple Support
• Connect a display to Mac mini - Apple Support

Use the Mac mini HDMI port and connect it to your monitor’s port

HP Support – HP25er 25-inch Display

That's a 25" monitor with a resolution of 1920x1080 pixels and a refresh rate of 60 Hz. It has two HDMI inputs and a VGA input.

Mac mini (Late 2014) - Technical Specifications - Apple Support

The Mac mini (Late 2014) has a HDMI port, and two Thunderbolt 2 ports with Mini DisplayPort connectors that can double as Mini DisplayPort ports. It supports "up to two displays at 2560 by 1600 pixels, both at millions of colors."

If the HP25er is going to be the Mac mini's only monitor, the easiest and best way to make the connection would be with a HDMI to HDMI cable. If the Mac mini already has one monitor, and you are using the HDMI port to connect it, then to connect the HP25er as a second monitor you'd want a Mini DisplayPort to HDMI adapter or adapter cable. I don't know if you would need an active adapter or if any old adapter would do.