Clause 1. Each House shall be the Judge of the Elections, Returns and Qualifications of its own Members, and a Majority of each shall constitute a Quorum to do Business; but a smaller Number may adjourn from day to day, and may be authorized to compel the Attendance of absent Members, in such Manner, and under such Penalties as each House may provide.
This clause has two separate and seemingly unrelated parts. First, the House and the Senate are given the power to judge the qualifications of their own members. In the case of a disputed Senate election, for example, in which both candidates claim to have won the vote, it is ultimately up to the Senate itself to decide which candidate gets the seat. Second, a majority of either chamber's membership is required to be present to constitute a quorum. Congress can continue to conduct business with less than a quorum present, but any member can then issue a "quorum call," requiring either that a majority of the members actually show up or that the house takes a temporary adjournment.
Clause 2. Each House may determine the Rules of its Proceedings, punish its Members for disorderly Behaviour, and, with the Concurrence of two thirds, expel a Member.
The House and the Senate have the power to set their own rules of parliamentary procedure. Over the course of 200+ years of American history, those rules have grown quite complex. The House and the Senate also have the power to kick out their own members for bad acts; expulsion requires a two-thirds vote.
Clause 3. Each House shall keep a Journal of its Proceedings, and from time to time publish the same, excepting such Parts as may in their Judgment require Secrecy; and the Yeas and Nays of the Members of either House on any question shall, at the Desire of one fifth of those Present, be entered on the Journal.
Both chambers of Congress must publish an official record of their proceedings. The Congressional Record is published daily while either house is in session, documenting all official activity on the floor of either house.
Clause 4. Neither House, during the Session of Congress, shall, without the Consent of the other, adjourn for more than three days, nor to any other Place than that in which the two Houses shall be sitting.
Neither the House nor the Senate can go out on extended vacation while the other remains in business, unless the other chamber approves. The idea here is to prevent one house from obstructing the other's legislation simply by refusing to show up to work.