I found a very simple circuit that delays the switching on of an LED.
With the given value of capacitance, there is a 0.7 second delay between the first and the second LED(4.7 micro farad).
(The image of the circuit diagram is in the attachment)
However, I want to use many LED's in parallel, instead of just one LED (I'm trying to write my name with LEDs where each letter has a certain delay between the previous letter). It didn't work when I just put more LEDs in parallel in the existing circuit. What are the necessary changes I should make in the circuit for it to allow many LEDs in parallel? I'm not well versed with electronic components. In fact I just barely understood this circuit. It would be great if I got some help with this! 
Ash_vin:
It didn't work when I just put more LEDs in parallel in the existing circuit.
What do you mean? How exactly did you connect them? What result have you got?
You should not put LEDs in parallel; each one will need its own series resistor.
Also, the 100 Ohm series resistor will result in a significant current of around 100 mA (depending on the Vf of the LED) which can possibly damage the LED. The BC557 is a small signal transistor and is rated for an Ic of 100 mA.
So for that reason you can not parallel leds plus resistors and drive it with one transistor.
You will need (for each LED) a BC557s, a 220 Ohm resistors for the base and led plus 100 Ohm resistor (basically copying the right hand side of your schematic)
You can connect all 220 Ohm resistors to the junction of the 4.7 uF capacitor and the 1M resistor similar as the single 220 Ohm resistor in the schematic.
I'm not sure of the effect of multiple circuits connected to the junction of the 1M resistor and the 4.7uF; it might effect the timing.
If you want every letter of your name to show with a delay after one came up, you need to make more modification; I'm not sure how I would do that (rusty knowledge, would probably be trial and error 
from my side).
I would reproduce the left hand circuit for each LED but use a bigger capacitor for each successive circuit. I would double the capacitor on each section, but then I only have a short name.
If you have a longer name then look into using a NE555 chip as a monostable for each section. Trigger them all at once and again have the time delays increasing. You can use a resistor as well as a capacitor to make the delays longer so you have much more flexibility about what capacitors you use.
How many letters in your name ?
Grumpy_Mike:
, but then I only have a short name.
Grumpy_Mike is pretty long.
I only use my second name.
I only use my second name.
So the waitress doesn't greet you with "Good evening Mr. Grumpy !" when you go to a restaurant ?
I'd use a '328P board and a Max7219, then you could easily control 64 LEDs with whatever delay you wanted between them, and have dimmable also.
This breakout board I offer makes that easy.
http://www.crossroadsfencing.com/BobuinoRev17/
Video here
url=http://[https://youtu.be/DbnQFv-lwNg]https://youtu.be/DbnQFv-lwNg
I don't think you're going to find a better solution than that.
raschemmel:
So the waitress doesn't greet you with "Good evening Mr. Grumpy !" when you go to a restaurant ?
I am not normally on first name terms with serving staff, are you?
I think the OP doesn't want to use an Arduino.
Like this, no processors involved.
https://twitter.com/i/status/1319976311381778433
This is how he made it.
"I am not normally on first name terms with serving staff, are you?"
Oh yeah! Every Friday I go pick up the sushi and
they say " Hi Robert, same thing ?"
Every weekday I go to the Thai restaurant and they say
"Hi Robert, PhadThai again ?"
You get the idea...
WOW ! What a cool electric sculpture !
What current and voltage do you need for the LEDs? How many letters?
Do you want to use simple ICs like 555?
Are you limited with space/power/money/parts availability?
What effect do you want? Turn on 1st letter, than add second and so on until all are lit and they should stay on as long as power is applied? Or only single letter turned on and the previous turns off when the new on is activated?
Edit: the circuit from OP does not work as described. It turns on the LED for limited amount of time just after the power is applied and turn it off after some time.
What are the necessary changes I should make in the circuit for it to allow many LEDs in parallel?
For starters, each led should have it's own current limiting resistor.
Second, you have not posted a drawing of the circuit that failed.
Third, the type of leds used, their forward voltage, and the number of leds is required.
Fourth, as ALREADY pointed out by another poster, 100 ohms is TOO SMALL for a 12V power supply.
You may not be well versed in electronics, so here it is simplified:
TOO MUCH CURRENT (12/100=0.12A (120mA) , BAD !
If you are going to use electronic components (that you are not well versed in..), you NEED to know
SOMETHING about them,
i.e.
- Led Vf
- Ohm's Law (V=IR, P=IV) (The classroom version of this taught in college uses the symbol "E" (for Electromotive
Force, instead of "V" for voltage. Hence, the formulas become E = IR, P=IE , so the penalty for forgetting Ohm's
Law is a PIE in the EIR, get it ?)
So to calculate the led current you need to FIRST know WHAT kind of led and what the forward voltage is (Vf)
which is given at some rated current , like 20mA,. Once you know the forward voltage, you can calculate the value
of the current limiting resistor:
Let Vcc = 12V
Let Vf= 2V@20mA
.'.
RCL = (12V-2V)/0.020A = 500 ohms (NOT 100 ohms !) | Where "CL = Current Limiting"
( @ Smajdalf, the "!" is to prevent the OP from frying his led)