Today is June 18th, Muiriel's birthday!

Literal

Today [topic-は] June 18 [conjunctive-で], Muiriel [genitive-の] birthday [copula-です]!

The で here is the te-form of だ/です — Japanese's way of saying 'X is A and Y' without a separate conjunction. So '今日は6月18日で、誕生日です' literally chains 'Today is June 18 and is a birthday.' Note the use of half-width Latin numerals embedded in Japanese text — common online but full-width forms (6月18日) appear in this sentence.