Firstly: 

Offering the sacrifice (udhiyah) is prescribed and recommended, and it is a confirmed Sunnah or obligatory, according to differences of opinion among the fuqaha’. Please see the answer to question no. 36432. 

Secondly: 

One sheep is sufficient on behalf of a man and the members of his household, even if they are many, because of the report narrated by at-Tirmidhi (1505) and Ibn Maajah (3147) from ‘Ata’ ibn Yasaar, who said: I asked Abu Ayyoob al-Ansaari: How were sacrifices offered at the time of the Messenger of Allah (blessings and peace of Allah be upon him)? He said: At the time of the Prophet (blessings and peace of Allah be upon him), a man would sacrifice a sheep on behalf of himself and the members of his household, and they would eat some of it and give some to others. Then people started to compete and it became as you see (nowadays).. 

An-Nawawi (may Allah have mercy on him) said: One sheep is acceptable on behalf of one person and is not enough on behalf of more than one, but if a person sacrifices it on behalf of the members of a household, he has performed the ritual on behalf of all of them, and the sacrifice in their case is Sunnah kifaayah (i.e., if one has done it, it suffices for all).

End quote from al-Majmoo‘ (8/370). 

If he slaughters more than one, there is nothing wrong with that, so long as it is not for the purpose of showing off. 

Shaykh Ibn Baaz (may Allah have mercy on him) was asked:

Has Islam stipulated the number of sacrifices to be offered by a Muslim on the Day of Eid al-Adha, and what is the number if there is a specific number? 

He replied:

There is no set limit for that. The Prophet (blessings and peace of Allah be upon him) used to sacrifice two sheep – one on behalf of himself and the members of his household, and the other on behalf of those who affirm Allah’s Oneness among the ummah of Muhammad (blessings and peace of Allah be upon him). So if a person sacrifices one or two, or more than that, there is nothing wrong with it. Abu Ayyoob al-Anaasi (may Allah be pleased with him) said: At the time of the Prophet (blessings and peace of Allah be upon him) we would sacrifice one sheep, as we would eat from it and feed others. Then people began to show off by offering more after that. The point is that one sheep is sufficient if a person sacrifices one sheep in his house on behalf of himself and the members of his household, and he will have fulfilled the Sunnah by doing that. But if he sacrifices more than that – two or three or four – or he sacrifices a camel or a cow, there is nothing wrong with that.

End quote from the Shaykh’s website:

http://www.binbaz.org.sa/mat/11662

But it is better and is preferable to limit it to one sheep on behalf of oneself and the members of one’s household, because this was the practice of the Prophet (blessings and peace of Allah be upon him). 

It was narrated that Jaabir ibn ‘Abdullah said: I was present with the Messenger of Allah (blessings and peace of Allah be upon him) on the day of Eid al-Adha in the prayer place. When he had finished his khutbah, he came down from his minbar. A ram was brought to him and the Messenger of Allah (blessings and peace of Allah be upon him) slaughtered it with his own hand, saying “Bismillah wa Allahu akbar (in the name of Allah and Allah is Most Great), this is on behalf of myself and on behalf of anyone among my ummah who has not offered a sacrifice.”

Narrated by Abu Dawood (2810); classed as saheeh by Shaykh al-Albaani in Saheeh Abi Dawood. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

Undoubtedly adhering to the Sunnah is better than not doing so. if we say that the Sunnah is for the members of a household to limit themselves to offering one sheep as a sacrifice, to be done by the head of the household, that does not mean that if they sacrificed more than one they are sinning. They are not sinning, but adhering to the Sunnah is better than doing a lot of something. Allah, may He be glorified and exalted, says (interpretation of the meaning): “that He may test you which of you is best in deed” [al-Mulk 67:2]. Hence  when the Prophet (blessings and peace of Allah be upon him) sent two men on a mission, and they did not find any water, so they did tayammum and prayed, then they found water, one of them did wudoo’ and repeated his prayer, and the other did not. He mentioned that to the Messenger of Allah (blessings and peace of Allah be upon him) and he said to the one who did not repeat his wudoo’ and prayer: “You followed the Sunnah”; and he said to the other one: “You will have a twofold reward.” Which of them was better? It was the one who followed the Sunnah, even though the other would have a twofold reward, because he will only have a twofold reward because he did the deed twice, intending in both cases to draw closer to Allah, may He be glorified and exalted. So he will have the reward for doing two deeds, but he is not like the one who followed the Sunnah.

End quote from Fataawa Noor ‘ala ad-Darb. 

And Allah knows best.