The Talmud in Pseachim 50a says
והיה ה' למלך על כל הארץ ביום ההוא יהיה ה' אחד ושמו אחד ... ושמו אחד מאי אחד אטו האידנא לאו שמו אחד הוא א"ר נחמן בר יצחק לא כעולם הזה העולם הבא העולם הזה נכתב ביו"ד ה"י ונקרא באל"ף דל"ת אבל לעולם הבא כולו אחד נקרא ביו"ד ה"י ונכתב ביו"ד ה"י סבר רבא למדרשה בפירקא א"ל ההוא סבא לעלם כתיב ר' אבינא רמי כתיב (שמות ג, טו) זה שמי לעלם וזה זכרי לדור דור אמר הקב"ה לא כשאני נכתב אני נקרא נכתב אני ביו"ד ה"א ונקרא אני באל"ף דל"ת
"And G-d will be for a king on the whole land. On that day G-d will be One and His Name will be One. (Zech. 14:9)"..."And His Name will be One" And now is His Name not One?
Rav Nachman bar Yitzchok said, 'This World is not like the World to Come. In This World [His Name is] written with יו"ד ה"י and is called with אל"ף דל"ת. But in the World to Come it is all one: written with יו"ד ה"י and called with יו"ד ה"י.'
Rava thought to expound on it at a sermon. A certain man said to him, 'it is written "לעלם"'
Rav Avina asked about a contradiction, 'It is written "This is My Name forever (Ex. 3:15)" [and it is written] "this is my Remembrance for generations (ibid)"? The Holy One Blessed be He said, 'Not the way I am written, am I called. I am written by יו"ד ה"י and I am called by אל"ף דל"ת'
Rashi explains on this passage:
לעלם כתיב - זה שמי לעלם חסר וי"ו לשון העלמה
'it is written "לעלם" - [When it says] 'this is My name forever (לעולם)' it is lack a וי"ו, a meaning of 'hidden'.
כתיב זה שמי לעלם - בהעלמה וכתיב זה זכרי דמשמע קרייה ולא העלמה לשון אחר כיון דכתב זה שמי פשיטא דזה זכרי היאך יקרא אלא בשמו:
It is written "This is My Name forever - With hiding. And it is written 'this is My Remembrance' which implies [actual] calling and not hiding. Another explanation: Since it says 'this is My Name' obviously 'this is my Remembrance'. How else would He be called except by His Name?
To paraphrase:
The Talmud asks about the verse in Zech. 14:9 that says that in the Messianic Age G-d will be One and His Name would be One. The implication being that now, G-d's and His Name are not one. The explanation the Talmud gives is that today we are meant to call G-d's Name as Ad-nai (אדנ-י) when we are referring to the Tetragrammaton. However in the future, we will call G-d's Name by the way it is spelled.
The Talmud then mentions a story where a person teaches a Rabbi, that when it says "this is My Name forever" the word forever is spelled missing a letter to allude to the alternate meaning of that root - to hide. And the intent of the verse is to teach that this Name is meant to be hidden, not pronounced.
Lastly - following the theme of the passage, a Rabbi points out a contradiction within the verse of Ex. 3:15. Rashi provides two interpretations to the contradiction:
The first part of the verse is meant to teach us to hide and not utter G-d's Name, but the second part of the verse uses the word "Remembrance (זכר)". This word also has a connotation of 'calling' and indicates that G-d's Name is meant to be called this way.
Once the verse said, "this is My Name", obviously we are meant to call G-d by this name and the words "this is my Remembrance" is superfluous.
The Rabbi answers that its teaching us that the way we are meant to write G-d's Name is not the same as the way we are meant to say it.
In practice we are taught in Shulchan Aruch O"C 5:1
יכוין בברכות פירוש המלות. כשיזכיר השם, יכוין פירוש קריאתו באדנות שהוא אדון הכל, ויכוין בכתיבתו ביו"ד ה"א שהיה והוה ויהיה.
Have intent of the meaning of the words when making blessings. When mentioning the Tetragrammaton: intend its read form of 'Ad-nai' that He is the Master of all. And intend its written form of 'יו"ד ה"א' that He was He is and He will be.