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I was informed that in a circuit, the current will stay the same, and this is why the lightbulbs will light up (because in order for the current to stay the same, the drift speed of the electrons need to get faster). However, I do not understand why the current needs to stay the same from point to point.

Why does the current stay the same from point to point in a circuit?

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When you sya "the current will stay the same," do you mean that it's the same from moment to moment (ie, constant in time), or that it's the same from point to point along the current? – Colin McFaul Feb 22 '13 at 3:40
welcome to physics.SE . I have given you a +1 so you can start accumulating some reputation that will allow you to comment. You can edit your question using the "edit" link above so as to clarify what Colin asked you.. – anna v Feb 22 '13 at 8:07
Sounds like chapter 19 of the third edition of Matter & Interactions by Chabay and Sherwood. It also sounds like you haven't read the chapter, because your question is addressed very early on in the reading material. Read it again VERY carefully, remembering that fundamentally, current is counting electrons drifting past a given point in a one second duration. – user11266 Mar 10 '13 at 22:55

The current doesn't necessarily remain constant in time in a circuit. For a simple example, consider a so-called $LC$-circuit. In this circuit consisting of an inductor and a capacitor in series, the current oscillates with angular frequency $$ \omega = \frac{1}{\sqrt{LC}} $$ were $L$ is the inductance of the inductor, and $C$ is the capacitance of the capacitor. In fact, the $LC$-circuit is a simple example of a more general class of circuits called AC or "alternating current" circuits where the current is a periodically changing function of time.

Having said this, there are many circuits in which the current doesn't change much once the system is in the steady state. A circuit consisting of a lightbulb (which is basically just a resistor) connected in series to a battery is a simple example. In this case the current $I$ satisfied Ohm's law and is given by $$ I = \frac{V}{R} $$ where $V$ is the voltage of the battery and $R$ is the resistance of the bulb.

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Electrons are present everywhere in an electric circuit. When a potential difference is applied to the circuit, an electric field is set up throughout the circuit, almost with the speed of light. Electrons in every part of the circuit begin to drift under the influence of this electric field and a current begins to flow in the circuit immediately.

You have to note here that if the potential difference you're applying is constant as with a D.C battery the electric field remains constant, and thus the current remains constant.

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The person with lowest number of  \$ $ \$  was the only one who wrote something on topic. Not so surprising for me… – Incnis Mrsi Oct 25 '14 at 7:43

The original poster explained unambiguously that the question is about spatial dependence, not temporal, i.e. a partial case of the Kirchhoff’s current law was interested about.

Tiny electrons are not things providing high-level integrity of electric circuits; they are only charge carriers. It is a good metaphor: when you send a mail, you do not care about carriers and their speed; you care about integrity of mail and terms of delivery. For an explanation how this integrity is ensured along a non-branching circuit read Why isn't resistance proportional to distance squared

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