Conjunction or birth of new moon is the same thing. “A conjunction occurs when the Moon passes between the Earth and the Sun; but because the Moon usually passes north or south of the Earth-Sun line, it specifically occurs when the Earth-Sun-Moon plane is perpendicular to the plane of Earth’s orbit around the sun.” This moment is called “New Moon Birth” or conjunction. At this moment the sunlight falling on the moon cannot come to the earth. In other words new moon is “No Moon”. No one can see this “New Moon” even by most powerful telescopes.
(Courtesy of Moonsighting.com)

Mufti Saeed Palunpuri mentioned the aayah of the quran, همصيلفرهشلامكنم دهشنمو (And whoever of you witnesses the month, let him fast it)

The word دهش means to witness something. Witnessing requires that a person observe something with their senses. If it is a thing which is spoken, the witnessing of it is with the ears. If it is something that is seen with the eyes, the witness to it must have seen it with his eyes. If it is a thing that is tasted, the witness to it must have tasted it with his own tongue. If it is something that is smelled, in order to witness it, a person must have smelled it with his nose. If it is something that is touched, in order to give witness of its feel, the person must have felt it.

You cannot witness that something smells a certain way, after only having looked at it with the eyes.

So now we are ordered to witness the month. And the question is, which of the five senses do we witness the month with?

Can you touch the month of ramadan? Can you hear the month of Ramadan arrive? Can you taste the month of Ramadan with your tongue? Can you smell a month with your nose? None of these 4 senses allow us to observe directly the month of Ramadan.

Rather, the witness of the month of Ramadan is to see with the eyes the moon of the first night of Ramadan. Only after seeing that, we could give testimony that we have seen that it is the month of Ramadan.

This meaning is confirmed, and none of the other meanings is even possible.

So from this single word in the Qur’aan, the entire debate is clearly and conclusively settled and two meanings are not even possible in the discussion.

Go after about 15 minutes after sunset. Look in the direction of the setting sun, just above it, also look to the right or left of it up to 30° in either direction of the setting sun. Keep looking until the time of moonset that you can obtain from local newspapers. Binocular helps. Have one or more persons with you, if possible. If you do this for a few months, you will know yourself about improving your procedure. What evening you should go to look, can be found from my web site, or if you know when was the moon sighted in your area for the previous month, then count 29 days from it.
(Courtesy of Moonsighting.com)

As most of us know, the moon goes through several phases. The months of the Islamic Calendar are based on sighting of the new moon every month. In Astronomy, a new moon means “when the moon of the previous month disappears (i.e. the moon goes from a slightly visible crescent to a completely black sphere that is impossible to be seen).” Remember, new moon is dark and invisible. About 17-23 hours after the new moon, a thin crescent becomes visible on earth; this is visible new moon. As we were taught by Allah (Qur’an 2:189) and the Prophet Muhammad (peace be upon him), it is the sighting of this crescent that marks the start of a new month. Therefore, even though a moon may be born on one day, it may not been seen until the next day. Furthermore, if a new moon is born less than 15 hours before sunset on day 1; it will not been seen until sunset of day 2; and when it is finally seen it is 15+24=39 hours old and will look very thick. This does NOT mean that this is a second day moon. A first day crescent-moon can be very thin in some locations and it can also be thick in other locations because of time difference. As Muslims, we should always remember that our first priority is to follow the teachings of our beloved Prophet (peace be upon him) who told us to start the month when we see the crescent, not when the moon is “born.” If we do that, Insha-Allah we will start the month at the “right” time (the time that Allah intends for the month to start for a specific location).
(Courtesy of Moonsighting.com)

Time passed after New Moon Birth is called the age of the moon. Sighting is possible at different age in different months. So, age cannot be a criterion for sighting. Why is it so? Because, the orbit of the moon is elliptical and in its orbit, the moon moves faster when it is closer to the earth, and slower when it is farther from earth. When it moves faster, the moon becomes visible at smaller age (like 17 hours), and when it moves slower, it becomes visible at larger age (like 23 hours). The main factor that makes the moon visible is the angle between moon-earth-sun. When this angle becomes about 9 degrees, the moon starts to be visible. How much time it takes to get this angle depends upon the speed of the moon in its orbit.
(Courtesy of Moonsighting.com)