What you have is Cat5 cable from each room punched down to what is called a 110 punch block. The white and blue single wire cables running back and forth over the top of the punched down wires are your phone line feeds to the wires to each room. Your house is wired for 3 phone lines (they may not all be used), one for each pair of the blue and white wires. The Cat5 cables have four pairs of wire, blue and blue/white, orange and orange/white, green and green/white, and brown and brown/white. You will see on the 110 block that the tops of certain posts between wires are colored blue, orange, green, and brown. For each pair, the white wire with the color stripe goes above the colored post, the solid color wire goes below. A set of 8 wire slots surrounding 4 posts starting from blue at the top to brown at the bottom makes up one wire set. Whoever initially installed your wiring did a nice job terminating the cables to the rooms so they probably fully meet the Cat5 standard up to the punch block.
To remove your room jack from the phone line carefully lift the blue and white/blue wires crisscrossing the jacks from the posts labelled for your room, leaving them connected to all of the other posts. A small crochet hook or similar can help lift the wires out. Carefully bend them out of the way.
To make your new connection easily order one of these: http://www.showmecables.com/product/3-FT-110-Style-to-RJ45-Patch-Cable-568B-Cat5e.aspx?utm_source=shopping+com&utm_medium=cpc&utm_campaign=ICPCSB03BL&utm_content=ICC&utm_term=Telephone_110+Patch+Cables&zmam=49733141&zmas=1&zmac=10&zmap=ICPCSB03BL
The flat end of this cable will push (hard!) over the pins for your room making a connection with the wires at the 110 block. Make sure the blue wires in the flat connector are on top. There should be a noticeable snap when the connector clicks in place. You may have to "thump" on it pretty hard. The other end goes into your switch.
An alternative would be to punch down a cable where the blue and white cables were removed and terminate it on an RJ45 jack, but that would require a punchdown tool and crimper that you probably don't have (a little knowledge and experience wouldn't hurt either).
You can probably find a professional or possibly an experienced friend to help you out if need be, but my company would charge you about $80 for me to drop in and spend less than 10 minutes to set you up. For that you can buy the cable and the switch to connect it all together and have a pizza to celebrate.