What type and size of needle does this machine use?

It sounds like there may be an issue with the needle clamp on your Necchi NC204D sewing machine. The fact that the needle falls out when you try to tighten it suggests that the clamp may not be holding the needle securely.
Here are a few steps you can try to fix the problem:

1. Check the needle clamp: Make sure that the needle clamp is tightened properly and that the screw is secure. Also, make sure that the needle is inserted all the way into the clamp.
2. Check the needle: Make sure that you are using the correct type and size of needle for your machine. The needles that came with the machine should be the correct size, but it's good to double-check.
3. Check the needle plate: Make sure that the needle plate is clean and free of any debris or burrs that could prevent the needle from seating properly.
4. Check the needle alignment: Make sure that the needle is aligned properly and that the flat side of the needle is facing the back of the machine.
5. If the problem persists, it's best to contact Necchi customer service or a professional sewing machine technician for further assistance.

It's important to note that working with a sewing machine can be dangerous, if you're not comfortable working with it, it's best to get it serviced by a professional.

Those are pretty common needles. Just look for 130/705 on the package. Then you will need to buy the size 11 or 14.

https://www.sewingpartsonline.com/universal-needles-schmetz-5-pack.aspx?

Looks like this needle will fit your machine:

Needles Organ Type 15X1SP 10pk

Bernina 2500DCET special coverlock needles sewing discussion topic... according to this discussion, the 130/705H (or 15x1) needles work in your machine, so you don't need to be limited to a particular brand name.

Not sure about your particular machine, but some overlockers recommend needle size no larger than a #90/14.

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Your owner's manual should tell you what type needles your machine uses. According to this site, 130/705 needles will work with your machine, are also very common, and are available in several brands: Schmetz, Janome, Organ, etc.

http://www.sewingpartsonline.com/necchi-6018-sewing-machine-parts.aspx

In addition, you will need to determine the needle sizes (60/8 through 120/19) as well as the needle point (universal, ball-point, sharp/microtex, top stitch, embroidery, leather...) so the needle is compatible with the thread and fabric you are stitching.

SCHMETZ Household Needle Chart

Be sure to change the needles frequently. A needle should be replaced about every 8 hours of sewing time. Sewists tend to skimp in this regard (although needles are the least expensive fix!), but the needle is the most frequent cause of sewing machine issues. So, if your machine begins to misbehave, the first thing to try is a brand new needle.

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Needles are sold at WalMart or other stores. The needle size is depending on what you are going to sew. The needles have the sizes marked on the package. The type of needle that your machine uses is a standard needle for sewing machines; no special type.

The stem of the broken needle may have a number or letter of the type of point you had. I believe the stems are all the same size, or that it doesn't matter as most are secured with a thumb screw.

Here is an article about the tips of sewing needles and how to choose them:

http://www.sewmamasew.com/2009/06/sewing-machine-needles-how-to-choose-them/

What kind of thread are you using. The most often problem today is that the thread is too thick for the size of needle. Try using a Metallica size 90 or a top stitching needle size 90
if the thread is a cotton type or a quilting thread it is often to thick to travel easily through the eye of a needle quickly to form a good stitch.

Your problem has been posted against the Elna 2007 mechanical sewing machine - not sure if this is your machine but it shouldn't make much difference. All domestic sewing machines have very similar needle placement.

The needle bar is the piece that goes up and down and holds the needle. There is usually a housing at the bottom with a thumb screw that is used to secure the needle. To change, turn your flywheel towards you until the needle is at its highest point. Turn off the motor or unplug machine for safety. Hold the needle with one hand and undo the thumb screw with the other. Insert a new needle into the housing FLAT SHANK TO THE BACK, all the way up into the housing, then tighten the thumb screw.

Standard domestic sewing machine needles are coded 130/705H and almost all makes and models will use this type of needle. Industrial machines are different and some overlockers/sergers will use a slightly longer needle called ELX.

The important thing is to change the needle regularly, about 4-6 sewing hours is maximum before the point starts to go blunt. If you can hear a squeaking or popping sound as you stitch, this is most likely a blunt needle. Also use the right sized and type of needle for the work you are doing. So for knits and jerseys, use a ball point needle to avoid skipped stitches. For lycra use a stretch needle coated with blue, helps with this synthetic fibre.

For wovens, use a sharp or universal point but pick the right size for the weight of fabric. Sheers, organza and fine silks, use a 60. Lightweight cottons, use 70. Medium weight fabrics use a 80 etc, right up to 110 size for coated upholstery fabrics. There are special needles for denim too as this is a very densly woven fabric so again, a blue coating helps to get the needle through the fibres.

I trust this helps you to resolve your problem.

Schmetz make a blue coated needle, usually best for denim fabrics as the coating aids the needle in slipping between tightly woven fibres such as denim. It may also be called a Denim needle.

But you shouldn't need to use that needle for most fabrics, just the most appropriate sized needle and either sharp point for wovens or ball point for knits. Perhaps get a mixed pack of both sharps and ball points to start with.

Try this website if you want to shop for needles on line, http://www.sewingpartsonline.com/sewing-machine-needles.aspx they carry a wide range of types, sizes and brands. Klasse and Schmetz are both high quality needle brands. You want the 130/705H system of needles for your domestic sewing machine. These are the norm for 99% of domestic sewing machines.

There are other needle systems for industrial machines (round tops) and EL705 for overlockers, these are extra long but identical shank design at the top. So just make sure you get the regular needles 130/705H.

In terms of what size to use, here is good reference guide on sewing machine needles.
http://sewing.about.com/od/sewingmachineindex/ig/Sewing-Machine-Needles/Needle-Sizes.--94.htm
The important thing is to change the needle for each new project, ie about 4-6 sewing hours, as a blunt needle will never give good results. If you always start with a new one, and its the right sized needle for the fabric you are sewing, then you should ensure best possible stitching results from your machine.
If the needle is too big, it will leave holes in the fabric, too small and it will break. If you use a sharp point on stretch fabrics, you'll get skipped stitches. If you sew a lycra knit with a ballpoint and it still skips, then change to a "Stretch" needle, this will also be blue coated ballpoint and helps with skipping and needle deflection.

Then there is twin needles, twin stretch needles, winged needles and leather needles, all for special techniques. Never worth trying to scrimp on needles, I have probably have in excess of 40 packs of different sizes and types in my sewing cabinet.

Hope this helps you out.

and they sho

According to Euro-Pro.com
Where can I buy new needles?Answer: The needles are universal and can be purchased at any sewingmachine notion department. The size and type of needles should beappropriate to the fabric used. See the needles and fabrics chart inyour instruction book.

http://www.euro-pro.com/europro/faqdetail.aspx?modelno=9000
has all the information including a downloadable manual (pdf file)