Clause 1. The Senate of the United States shall be composed of two Senators from each State, chosen by the Legislature thereof, for six Years; and each Senator shall have one Vote.
Each state, regardless of population, gets two seats in the Senate. Each senator's term lasts six years. And note the crossed-out bit: originally, senators were not elected by the people but were instead chosen by their state legislatures. This led, by the end of the nineteenth century, to widespread accusations of corruption. The 17th Amendment, an important Progressive Era reform passed in 1913, allowed for the direct election of senators by the people.
Clause 2. Immediately after they shall be assembled in Consequence of the first Election, they shall be divided as equally as may be into three Classes. The Seats of the Senators of the first Class shall be vacated at the Expiration of the second Year, of the second Class at the Expiration of the fourth Year, and the third Class at the Expiration of the sixth Year, so that one third may be chosen every second Year; and if Vacancies happen by Resignation, or otherwise, during the Recess of the Legislature of any State, the Executive thereof may make temporary Appointments until the next Meeting of the Legislature, which shall then fill such Vacancies.
This isn't as complicated as it sounds. The idea is simply to ensure that one third of the Senate's seats are up for election every two years, making the Senate a "continuing body." That just means that all the senators will never face election at the same time. The deleted section at the end was overwritten by the The 17th Amendment, which allowed direct election of senators in 1913. The The 17th Amendment requires any Senate vacancy to be filled in a special election called by the state's governor.
Clause 3. No Person shall be a Senator who shall not have attained to the Age of thirty Years, and been nine Years a Citizen of the United States and who shall not, when elected, be an Inhabitant of that State for which he shall be chosen.
Basic job requirements for the Senate: You have to be at least 30 years old, a citizen for at least nine years, and a resident of the state you want to represent.
Clause 4. The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.
The vice president has a pretty useless role in the government… unless the president dies, of course. Otherwise, the VP's role is largely ceremonial; he gets the cool-sounding title "President of the Senate," but he doesn't actually get to participate in its debates or cast a vote, except in the rare circumstance of a tie vote. Most vice presidents thus opt not to bother even showing up in the Senate, except on special occasions.
Clause 5. The Senate shall chuse their other Officers, and also a President pro tempore, in the Absence of the Vice President, or when he shall exercise the Office of President of the United States.
The Senate's president pro tem—typically a senior senator of the majority party—gets to run the show when the VP of the United States isn't in the house… which, in practice, is almost always. The Senate also has the power to choose (or "chuse," if you prefer) its other officers.
Clause 6. The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried, the Chief Justice shall preside: And no Person shall be convicted without the Concurrence of two thirds of the Members present.
If the House of Representatives votes to impeach any civil officer, the Senate must serve as judge and jury. If two-thirds of the senators vote to convict, the impeached official is removed from office. Twice in American history, the House has brought impeachment charges against sitting presidents—Andrew Johnson in 1868, and Bill Clinton in 1998. In both cases, the Senate voted not to convict, and the impeached presidents served out their full terms of office.
Clause 7. Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of Honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.
The only punishments the Senate is allowed to mete out in impeachment cases are removal from office and banishment from future government service. However, the regular courts could also then seek to impose further criminal penalties for wrongdoing.