What is the Afghan wedding ceremony like in The Kite Runner?

There were several ceremonies leading up to the actual wedding of Amir and Soraya. First came the lafz ("the ceremony of 'giving word' ") at the general's home before several dozen guests, with Soraya making her appearance at the end. They skipped the traditional Shirini-kori ("the 'Eating of the Sweets' cermony") because of Baba's illness. Baba spent most of the money he had saved (more than $30,000) on the nika, the "swearing ceremony." The wedding couple dressed in traditional green, a symbol of "new beginnings." Many of the guests spoke before Amir and Soraya approached the stage, followed by "a procession of uncles, aunts and cousins." The Afghanistan "wedding song, ahesta boro, blared from the speakers." Then followed Ayena Masshaf, a ceremony involving mirrors in order for the couple to "gaze at each other's reflection." They danced the attan, a circle dance, before

... the party moved from the banquet hall to Baba's apartment... until the neighbors called the cops.  (Chapter 13)

You can find a detailed description in Chapter 13 of the novel.

What are the similarities and differences between the wedding in The Kite Runner and a traditional wedding?

Presumably, you are asking how Amir and Soraya's wedding differs from traditional Afghani weddings. Amir and Soraya go through some of the conventional rituals and rites, but because of Baba's imminent death, the process is sped up to allow his participation.

When Baba is diagnosed with terminal cancer and decides to forego treatment, Amir asks him if he will discuss Amir's marriage to Soraya with the Taheris, the first step in the formal engagement process. Amir tells his father, "I want you to go khastegari. I want you to ask General Taheri for his daughter's hand" (161). The Farsi term khastegari means suitor, and in this case, Amir is asking for Baba to make Amir's interest in Soraya official by collecting the families and having Soraya's family approve of the match. Baba immediately calls and sets a meeting time with General Taheri and his wife Jamila. Baba goes to the house. Amir describes this action as Baba performing "one last fatherly duty" (163). He calls to tell Amir that the General has accepted, but Soraya wants to speak to Amir first. She comes clean about her past, when she ran away with an Afghan man without being married. This is considered shameful in Afghani culture, and in Afghanistan, it's probable that Soraya would never have found a suitor. However, Amir will not refuse to marry her because of it. He feels that he cannot judge anyone else's past, given the sins he has committed against Hassan.

Next, Amir and Baba participate in "lafz, the ceremony of 'giving word'" (166). After this ritual, Amir says, "Soraya's family would have thrown the engagement party...Then an engagement period would have followed, which would have lasted a few months. Then the wedding..." (169). Tragically, "Baba didn't have months to live," so the process goes much quicker. The couple does abide by the tradition of "never [going] out alone together while preparations for the wedding proceeded" (169). On the wedding day, many of the rites associated with Afghani marriage are maintained. For example, Amir says, "We did the Ayena Masshaf, where they gave us a mirror and threw a veil over our heads, so we'd be alone to gaze at each other's reflection" (171). The food is a traditional Afghani wedding spread, and a party follows the ceremony.

Amir and Soraya try to follow as much of Afghani tradition as possible in their courtship, engagement, and wedding ceremonies; however, because of Baba's terminal cancer, they cannot afford to take much time to complete all of the traditional rites.