What do Christians mean by a "personal relationship" with Jesus? When Christians talk about having a "personal relationship with Jesus," what does this mean?
Christians believe that Jesus is alive, he knows us intimately, and is our best friend above all others. A personal relationship with him is exactly that: a relationship very much akin to  that we have with other close friends and family members. Often people focus on the majesty and "kingliness" side of Jesus, and in doing so are tempted to distance themselves from him by seeing him as some distant, untouchable, unreachable figure on a throne high above us; but to have a personal relationship with him means to accept our own position as co-heirs with him, to share our lives much more openly with him.

As a Brit, when I meet (earthly) royalty I dress up and put on my best manners, and in effect put on a show. I don't need to do that with Jesus - he's seen me naked, in my own home, he shares my ups and downs and secrets that I certainly wouldn't blurt out to the Queen if I met her. So while acknowledging that Jesus is King, I don't treat him in a stand-offish formal way because he's also my best friend. That's what it means to have a personal relationship with him.