I have replaced an outlet with 4 wires. The outlet works, but the switches and outlets down the line do not. What have I got wrong?

By 4 wires do you mean 2 wires entering the box from one side and 2 more exiting the other side? Are the wires just 2 black and two white, no gren or bare? With the receptacle removed only one black wire should be hot, use a tester to determine which one. Hot black goes on the yellow brass screw and white to the silver screw. There are two brass screws, the other black goes on the other brass screw. The second silver screw gets the other white wire. When finished turn the power back on and test. You should get 120 between any white and any black. Back to you for your results. When starting this work test between the black and white to insure power is actually off.

Yes, two whites and two blacks plus a ground. One black is hot. I put two blacks on the brass side and two whites on silver side.

Did the other outlets work properly before you replaced that first outlet? If so - are you sure that the metal "bridge" that connects the two screws on either side of the outlet are intact? Finally - if you have an outlet tester (the kind that has indicator lights to show the status of the wiring) - what does it show? You can get one for just a few bucks at your local hardware store.

Yes. Everything worked before I replaced the outlet. The new outlet that I installed has power, but some of the other outlets in the same circuit do not. Also some do. So, I'm assuming the one I replaced is in the middle of the circuit and the ones that don't work are at the end. Correct?

Have you worked on any other outlets, switches, etc ? Also test hot to ground at the other receptacles and tell us what you find.

have you worked on any other outlets, switches, etc ? also test hot to ground at the other receptacles and tell us what you find.

No. Just this one. Tested all outlets and switches "down the line" and there is no power.

Humor me here. This is a new duplex receptacle like this?
http://www.cesco.com/resources/078477/199671-ProductImageURL.jpg
See the little piece of brass between the two screws on the side? Is it still there on yours? On both sides? This has already been asked, just differently.
You need a voltage tester, sounds like you have one. Turn the circuit off and ease the receptacle out of the box and let the wire support it, then turn the circuit back on.
Test between each hot screw and the ground to see if they are both hot when the wire is hooked up, if one is not hot, the brass is not connecting between the two screws
IF you find voltage on BOTH of the hot screws, test for voltage between a hot screw and each white(neutral screw) that will tell you if the neutral is good where it leaves the box

Please tell me what you find

I've done all that and it's all correct. The outlet itself is fine. It's everything beyond the outlet on the same circuit that doesn't work (a switch that controls overhead light and two more outlets and a double switch which controls two separate outside lights). By the way, the two outlets that don't work are both "end of circut". That is: they both only have two wires going to them

I've done all that and it's all correct. The outlet itself is fine. It's everything beyond the outlet on the same circut that doesn't work (a switch that controls overhead light and two more outlets and a double switch which controls two separate outside lights). By the way, the two outlets that don't work are both "end of circut". That is: they both only have two wires going to them

The fact that the replaced one is working and sending power out/downstream tells you that the problem is at another location,
You now need to check power in the same manner that I just indicated at the nonfunctioning locations.First check all the connections, wirenuts included, with the circuit turned off, before applying power and checking it hot.
If you find no power at all in the nonfunctioning devices' boxes, the next step is to troubleshoot the remaining boxes on the same circuit that ARE functioning,
When checking wirenuts in junction boxes, (circuit off) test them by tugging on each wire or disassembling the junctions one at a time and carefully re-wirenutting them, sooner or later the problem will reveal itself.

Before you start, The best way to make up receptacles in a professional manner is NOT to use the receptacle as a wire connector. Take the incoming hot and all outgoing wires that need to be energized as they go to the next device ( not switched wires going from a switch to a light fixture for example, that is a switch leg) and wire nut them together with one more short piece of wire that will go to the device. This is called "tailing out" the wire. Do the same with the neutrals (white) and usually the ground wires are already done this way or with a crimp sleeve. Tuck the wire nuts into the back of the box, now you have much less wire to force into the box with the outlet.
Done this way, the entire load downstream is not carried by that little piece of brass between the screws, I never put more than one wire to either screw on receptacles.

Avoid push in connections like the plague. Those devices are the absolute cheapest c##p sold.

Doubling and tripling wires into the back of a receptacle is done a lot in residential jobs because the profit margin is so small anymore, every effort is made to reduce the time it takes to get a job done, and wirenuts aren't free either. This is mechanically better, I like Ideal 341 and 452 wirenuts for most makeup in junction boxes, you might want a couple of grey, or blue ones (bigger) occasionally.