Here are layman's terms for some of these circumstances. Understand that these situations are a result of how the baby develops in utero, and are not a reflection of anything wrong with the baby or his parents. In some cases, this kind of development is genetic.
Thank God we live in a time in which highly skilled surgeons can make a relatively easy job of fixing the baby's genitalia, so he will not only have a beautiful circumcision, but in cases where certain elements of functionality would have been a little off, the correction is nothing short of miraculous. In a different time, some of these babies would not grow up to become fathers.
1. No foreskin
If a baby develops without any skin covering the glans, there is nothing to remove. The only procedure is Hatafat Dam Brit - drawing a minimal amount of blood from where the foreskin would have been, just above where the glans begins (not from on the glans itself).
This is the condition in which the hole of the penis - the urethra - is not where it should be. It could be a little off center on the tip of the glans. It could be below the glans. Or in more extreme (and very rare) cases, it could be anywhere on the shaft or even the scrotum. Depending on the severity of the case, reconstructive surgery would align everything properly and allow for a productive future. If the hole is on the shaft or the scrotum, a boy would always have to sit when going to the bathroom, and he would never be able to be a father. (If it's only a little off center, most surgeons would recommended leaving it alone).
In the event that surgery would be required here, the surgeon needs as much skin as necessary. If the plan is for there to be a circumcised look afterwards, the surgeon will know this and will produce that result. But anything a mohel might do in a classic circumcision would do more harm than good for the baby.
[In most surgery-required-hypospadias cases I've seen, the hole is at the bottom of the glans, at the spot where the frenulum is.]
The natural position the penis takes when erect is upward. In simple terms, chordee prevents that, as internal tissue pulls the erection downward, and even makes a downward curvature.
THERE ARE OTHER CONDITIONS AS WELL, BUT THESE ARE THE MOST COMMON
So What Is Done With the Bris?
There is no bris. The circumcision will be taken care of during surgery. The best route is to find a Jewish doctor (ideally observant), who will say the proper bracha (blessing) and have in mind the fulfillment of the mitzvah. If the surgery is not done by a Jewish doctor, the circumcision can be turned into a "Bris" through Hatafat Dam Brit, after everything has healed.
What About Family and Friends Who Are Looking to Attend a Bris?
They should be understanding that the baby's best interests do not include a formal bris. They don't need to know all the details. Some people are embarrassed to discuss the baby's circumstance. But it's really nothing to be embarrassed about. No one has done anything wrong. And the baby will be fine. And everything will work properly after the surgery.
Parents will do well to inform their family and friends something along the following lines: