What's the easiest way to relieve oneself in a car?

A well prepared fighter pilot driver will be equipped with a Urine Collection Device (UCD)

A typical UCD consists of a small container with a dehydrated sponge inside, connected to a tube which in turn is connected to a funnel-like orifice that is adapted to the user's anatomy (different designs are used for men as compared to women). The user simply holds the funnel near or on his penis or her vulva, and urinates into the tube, with the collected urine saturating the sponge (which may be impregnated with disinfectants and odor-control substances) and filling the container. UCDs are designed to be used in cramped quarters without requiring that the user rise from his seated position in the cockpit car.

The ultimate in such devices can cost up to $2000 and are available in male and female versions. Hanging on can involve real dangers:

Urinary Retention reduces attention span and the ability to make decisions to the same degree associated with alcohol intoxication or 24 hours of sleep deprivation.

Male Kit:

Female kit:

Of course this could be a little too expensive for the non-professional traveller, so home made methods are possible:

As @GayotFow suggests, a simple plastic bottle may also be used. Easy enough for male urination, but for female urination, a female urination device might work too. Many are available on ebay such as this one which look like this:

Or a slightly different type:

http://gottatinkle.com

If getting out of the car is allowed, then, for modesty, I recommend opening the front and back door away from traffic and relieving yourself between the two.

Honestly, adult incontinence products are inexpensive, highly effective, comfortable and hygienic. For example, a packet of 15 iD Slip Maxi all-in-one briefs (essentially adult diapers) costs £10 (66 p each), holds about 3-4 litres of urine, is dermatologically tested, safe, discrete, and comfortable. Glider and fighter pilots use such products, as do astronauts (where they are given the wonderfully euphemistic name 'maximum absorbency garments') as well as people with medical problems. It might not be dignified, but you don't need to actually do anything different to go to the toilet, and don't need to worry about being arrested.

An English male friend of mine always has several small milk bottles in the car, neck of which are big enough to allow them to be used as 'hospital bottles' and the tops are closing well enough to keep the contents in when done properly. Store them top up when used to make double sure.
Select the size of the milk bottles based on need.

Me, a female, personally prefer to get out of transport, use a bush where available but I could do with the blanket suggested by @Kate Gregory or the open doors by @Berwyn if nothing better is available.

And of course, always use the facilities where available, even when you do not feel the need yet, (good thing to train your youngsters on as well,) which will take care of the first few minutes of each leg of travel.
When you see a possible stop coming up, use your reasoning abilities or check with your passengers and pull in if there is the smallest option of need.

Find the nearest place to stop safely and get out of the car... Having traveled a LOT by car, including with/as small children, that's always been an option.
If someone has such serious problems holding their bladder, invest in diapers (which exist for adults as well).

My ex-husband drove a work truck for 2 1/2 hrs each way to the job site on the Los Angeles freeway system every day while drinking a thermos of coffee. One day he showed me his method. Drilled a hole through the floor board, stuck one end of rubber tubing into the hole, put a funnel on the other end. While driving he would "whip it out" and go into the funnel. Urine would just drain out right onto the freeway. I can only imagine how big the yellow brick road was between home and the financial district! It's gross, but effective and we had a great laugh when he got the nerve to show me how he did it.

You did not mentioned any specific country, so I adjust the answer accordingly. When I mention "yourself", it includes children, life partners and all other people fighting with the problem.

1. Be aware if it is ok to relieve oneself in public and simply leave the car when possible

Some cultures are more permissive and see the inevitability of the situation (meaning: They neither find it gross nor see it as punishable offense). Stop, go out (lock the car), find some bushes and relieve yourself. Tip for beginners: Urinate into a depression to avoid that the stuff goes on your shoes. For bigger business bury a small hole (using a stick if necessary), squat down over it and use handkerchiefs (or paper or leaves, but please, please look out for poison ivy/poison sumac....google it!).

2. Use some kind of container in the car

Naturally the problem occurs if you cannot stop, you are in not so permissive country and in a location without any cover. So with a rising level of despair you are looking for containers, any container. The best options are empty soft drink bottles or thermos jugs because they have the capacity, big enough openings and they can be closed to avoid the smell (and you need not to be afraid to accidentally knock it over). Simply put it in your lap, open it, place your genital against it and begin; people will only see something if they are standing right at your car. Women are admittedly at a disadvantage. Wheelchair drivers have often big, easy to handle bottles which can be bought in sanitary supply stores. They also have the utilities described by Berwyn.

If you are extremely desparate, one option is using as much nested plastic bages as possible and duck tape to close it down.

I must admit that I find condom urinals or diapers (except for small children) a bit of a overkill. The problem remains that you very likely need to change the underwear (because it is a rare problem and you won't use the stuff all the time...?) and a car has not much room for inconspicous anatomical maneuvering.

Disposable Portable Urinal

I use Travel John for tent camping and have one in my car for emergencies.

It's very small when folded, the bag is filled with a powder that turns into a gel so you won't spill anything. You can also add more to it later, the gel will absorb more liquids as well. The powder has some chemicals in it to neutralize the smell. Works both for ladies and gentlemen. It's biodegradable, once you are done with it, dispose in a garbage or throw out of the window :)

I had this case once when driving with my children back from vacation. By driving I mean sitting in the middle lane of a huge traffic jam.

Obviously, this was precise moment when the younger one absolutely had to pee (even though I asked 7 times if he was done at the last stop. But I digress).

The solution I found is to have him "do it" into a box of tissues. They absorbed the liquid fine. Would do again if faced with the same issue.

Ok, this seems like a silly question, but I will give it a try.

In the US, if your not on the interstate system, then you likely won't have this problem. There are few "traveled" roads, that don't have small towns and what not along the path. They do exist, specially in the rocky mountain regions. Adjust the answer as needed.

Number 1

Pull off an exit, pull back on the entrance ramp. As you go up the entrance ramp pull way off the ramp (your more likely to be seen if your on the entrance ramp then the exit ramp, also cars are generally going slower). Open the Passenger door. Stand at the passenger door facing the inside rear of the car.

Number 1 - Male

Whip it out. Point and shoot.

Number 1 - Female

Uncover. Let it flow.

Number 2

Don't. Just don't. Wait till the next stop. Even if the next stop is a ways away, you can hold your number 2 for a while (normally). There's generally no reason you can't make it to the next stop. (remember that on an interstate the furthest point between two rest areas is about 2 hours)

Can't stop - Won't stop

My grandfather was a professional truck driver. He always said, don't do the can't stop/won't stop method. There is no reason to risk your health just to save a few moments time. When you have to go, go. That said here are your answers.

Number 1

Don't be stupid. Driving with your wang out is a recipe for disaster (I don't imagen it's much different for gals. Doesn't really matter the circumstances. So lets assume your not driving.

You need a container that is large enough to hold all the urine. You will need a container that can hold a liter or more likely (for adults). You don't want to have to switch containers.

Number 1 - Male

If possible you want the container to be large enough to (sorry) "put the tip in" but that may not be possible. Your basically working with what you got. Like before point and shoot. If you can find a bottle with a large enough opening, rest "it" just inside the top, if not then line up the "entry and exit" holes. Just make sure that you have a large enough container.

Number 1 - Female

Ladies have it harder on this front for sure. First you need a funnel. A real funnel may be nice (if not coated in engine stuff, remember where your about to stick this thing). If that's not an option time to make a funnel. Cut off one corner of a large bag of chips. A plastic bag, will work as well. Cut one bottle in half and just it as a funnel into a second bottle. Just make sure the opening on the receiving bottle is larger. Scoot to the edge of the seat. Cover up as much of the "area" as you can with the funnel. Let it flow.

Number 2

Nope, just don't.

Super Best answer

Pay more attention to your body. Pee when you can, poop when you must. Every time you stop pee. Every time. Bring an ass gasket if that's your concern. Every time you get gas (300 miles is only 5 hours. That's a stretch for kids, easy for adults). If your traveling with kids plan on stopping every hour. If your Traveling with just you, or just adults, plan on stopping ever three. Do a pee check at every stop. "Were not going anywhere till you pee mister. Now get in there and pee!" if you have to. A reasonable amount of planning is really all that's required. Again if your in the US and on the interstate it's rather difficult to get caught if you stop at every rest area and make sure everyone pees (again assuming everyone is healthy). If your taking the highways, then there will almost certainly be a small town, though there are a few areas that don't.

This solution works only for males.

Karl Pilkington proposes a solution in this video:

https://youtu.be/rt2h8jSdxHY

Store a hot water bottle under your passenger seat. When necessary, relieve yourself in that.

The advantages (from the video) are:

• large storage capacity
• wide rim to prevent, er, spillage