May the peace, mercy and blessing of Allah be upon you also.
In the name of Allah, the Beneficent, the Merciful,

All praise is for Allah, and may peace and blessing be upon the Messenger of Allah. To proceed:

If the good brother asking the question is referring to a man shaking the hand of his brother in faith, then this is a legitimate, recommended act in Islam.

As for a man shaking hands with a non-mahram woman, it is not permissible under any circumstances, whether the woman is young or old, married or single, a relative or non-relative, Muslim or non-Muslim, whether the hand-shaking is done with covered or uncovered hands, with desire or without; all of that is haram.

The evidence for that is plentiful. For example:

The hadith from `A`ishah (may Allah be pleased with her) who said, "The hand of the Messenger of Allah (may Allah bless him and grant him peace) never touched a woman`s hand except a woman he owned [his captives and female slaves]." [An authentic hadith related by al-Bukhari]

On the authority of Umaymah bint Raqiqah (may Allah be pleased with her), the Messenger of Allah (may Allah bless him and grant him peace) said, "Verily, I do not shake hands with women." [An authentic hadith related by Ahmad, at-Tirmidhi, an-Nasa`i, Ibn Majah and others]

According to another narration, he said, "I do not touch women`s hands." [An authentic hadith related by at-Tabarani, declared authentic by al-Albani]

No one claims that this is only for the Messenger of Allah (may Allah bless him and grant him peace) because the basic principle is that no verdict in Islamic law is particular to the Messenger of Allah (may Allah bless him and grant him peace) until it has been established with proof that the ruling is only for him.

There is no authentic proof that the prohibition of shaking hands with non-mahram women is particular to the Messenger of Allah (may Allah bless him and grant him peace). One proof of this is the following hadith which contains a severe warning for anyone who touches a non-mahram woman:

On the authority of Ma`qal ibn Yassar (may Allah be pleased with him), the Messenger of Allah (may Allah bless him and grant him peace) said, "That one of you be speared through the head with a needle made of iron would be better for him than to touch a woman who is not permissible for him." [An authentic hadith related by at-Tabarani and al-Baihaqi, and declared authentic by al-Albani]

Shaikh al-Albani (may Allah have mercy on him) said, "In the hadith there is a severe warning for anyone who touches a woman who is not permissible for him, which is proof that it is haram (forbidden) to shake hands with women, for that is undoubtedly one of the meanings contained in the word, "touch". Many of the Muslims have been brought to test with this [issue] in this age."

Allah—the Glorious and Exalted—has said (in meaning), {Tell the believing men to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts, etc.). That is purer for them. Verily, Allah is All-Aware of what they do.}[An-Nur 24:30] Allah Almighty has also said (in meaning), {And tell the believing women to lower their gaze (from looking at forbidden things), and to protect their private parts (from illegal sexual acts, etc.)…}[An-Nur 24:31]

Shaikh Shanqiti (may Allah have mercy on him) said, "Verily, Allah ordered [us] to lower [our] gaze for fear of falling into temptation, and without a doubt, one body touching another body is stronger in arousing the sexual instincts and a stronger call to temptation than looking with the eye. Every fair person knows that to be true."

The imams of the four schools of Islamic jurisprudence and other scholars agreed that it is haram for a man to shake hands with a non-mahram woman.

Imam as-Safarini (may Allah have mercy on him) said, "This is the opinion of the four imams and the scholars in general, and may Allah have mercy on them all."

There is no truth to what is said about `Umar ibn al-Khattab, that he used to shake hands with women. That is a lie about `Umar (may Allah be pleased with him).

And Allah Almighty is more exalted and more knowledgeable.