The article seems quite confusing: the text and figures don't match. I'll try to present here the same three schematics as there, with hopefully a more matching explanation.
Assume U1 is your microcontroller, and P1 is an I/O pin configured as input. (It could be any logic gate, really.) Other connections to U1 are not that relevant so are not pictured, but assume it has power connections and other necessities.
(1) If the button is pressed, port P1 is connected to ground, and will sense a low logic level. But when the button is released, the port isn't connected anywhere, but is floating. There's no definite voltage present, so even minor noise may cause the digital input to switch from one value to the other. It might also oscillate, and cause increased power consumption. Not good.
(2) Now, when the button is not pressed, the port will sense a high level, since it's connected directly to Vcc. But if the button is pressed, Vcc is short-circuited to ground, and the power source will probably burn and die. Even worse.
(3) Here, if the button is not pressed, the port will again sense a high logic level: it's pulled high through the resistor. (There's no voltage loss over the resistor, since the impedance of the digital input is very high, and therefore the current to the port is approximately zero.)
When the button is pressed, the port is connected directly to ground, so it senses a low level. Now, a current will flow from Vcc to ground, but the resistor will limit it to something sensible. This is good.
In this schematic, an unpressed button reads as a high value (1), and a pressed button reads as low (0). This is called active-low logic.
Swapping the resistor and the switch would invert this, so that an unpressed button would read as low (0), and a pressed button as high (1). (active-high logic.)
simulate this circuit – Schematic created using CircuitLab