Why did Emperor Nero kill his mother and others?

Nero's mother always wanted to get power. Note that she was married to emperor Claudius who already had a son. However, Agrippina convinced Claudius to actually allow her son (Nero) to be the emperor.

After that Claudius died in obscure circumstances.

At that time Nero was really a kid, so Agrippina actually had a lot of power. When Nero became an adult, Agrippina had to effectively let that power go, but I guess that is difficult for someone used to always using and enjoying that power.

If, on top of that, you add the character of Nero, you have the perfect c**ktail.

SHORT ANSWER

Nero, in common with many other rulers, killed people he thought were a threat to him (his mother, his half-brother, those involved in the Pisonian conspiracy). His reason for persecuting the Christians is disputed - they just have been an easy target after the Great Fire of Rome.

LONG ANSWER

Agrippina and Britannicus

Nero's mother (Agrippina) was both ambitious and ruthless. With Claudius (her husband) out of the way

Agrippina now meant to rule through her son....Her power was advertised on the coinage which bore confronting busts of herself and Nero on the obverse....Nero’s name and titles were banished to the reverse.

Source: H.H. Scullard, 'From the Gracchi to Nero'

It would hardly be surprising if Nero, as Emperor, had resented this. Although he at first accepted (or at least tolerated) her dominance, he eventually worked to undermine her power, at which point she turned to Nero's half-brother Britannicus:

she attempted to bring Nero back into line by threatening to champion the cause of Britannicus. Claudius’ natural and adopted sons had never enjoyed a good relationship; Britannicus’ criticism of Nero’s singing voice and his reference to his adoptive brother by his original name of Lucius Domitius can hardly have been harmless banter.

Source: David Shotter, 'Nero'

Nero wouldn't have to have been paranoid to believe that his mother was planning to dispose of him, perhaps in favour of Britannicus (which made him a threat as well). Nero and Agrippina's relationship deteriorated further, as related by Shotter:

Angry recriminations between mother and son led to her expulsion from the imperial presence, and to her ill-judged fostering of other friendships designed to aggravate her son....Increasingly Nero identified his mother as the one principally determined to check his pleasures and to interfere in his life. Things took a far more serious turn when, probably in ad 58, Nero began his love affair with Poppaea Sabina, a lady whose noble lineage and expectations were in a class very different from those of Acte. It was Agrippina’s opposition to this, and Nero’s determined desire to be free to ‘lead his own life’, that convinced him that his only solution was to rid himself permanently of his mother.

The Pisonian Conspiracy

Although Nero could never be considered to be anything but a tyrant (though his early years showed some promise), his brutality was hardly unusual in the context of the deadly politics of Imperial Rome. This brutality was most evident after the AD 65 plot to kill him was exposed.

Nero took savage revenge: trials intra cubiculum principis were revived, and his senatorial victims included Piso, Seneca and his nephew the poet Lucan. In the first flush some nineteen persons including Faenius were killed and thirteen exiled...

Frightened by the narrowness of his escape Nero became a ruthless tyrant, employing more spies and the surviving Praetorian Prefect, Tigellinus, to hunt down all suspects. His victims included the son of Ostorius Scapula, the former governor of Britain, and C. Petronius, Nero’s elegantiae arbiter

Source: Scullard

The Christians

When considering Nero's persecution of the Christians, it's important to recognize that Nero was only one of many emperors to do so. As to why,

Most likely Christ-believers were singled out because they were regarded as committing national apostasy. By abandoning and even critiquing Roman religion they were religiously impious and politically disloyal. While Roman religion was pluralistic, it was not necessarily tolerant towards foreign cults, especially if they were thought to promote debauchery and disorder. For a Roman resident to profess faith in Jesus as an alternative to Roman religion lent itself to accusations of atheism and hatred of the human race, and was interpreted as a rejection of the mos maiorum (ancestral customs) and committing maiestas (affronting the majesty of Caesar).

After the Great Fire of Rome, it has been argued, Nero needed a scapegoat and the Christians were an easy target.

Why did Nero blame the Christians? The answer may be that they were living near the place where the fire started: the eastern part of the Circus Maximus.