Women are allowed to pray in the mosque, although praying at home is better for them. Read the full answer based on the authentic Sunnah and the understanding of the Salaf.
All praise be to Allah, and peace and blessings be upon the Messenger of Allah (ﷺ).
It is permissible for women to pray in the mosque. This is clear from the following authentic hadith:
From ‘Abdullah ibn ‘Umar (رضي الله عنه), who said: "I heard the Messenger of Allah (ﷺ) say: ‘Do not prevent your women from going to the mosque if they ask your permission.’" Bilal ibn ‘Abdullah (the son of ‘Abdullah ibn ‘Umar (رضي الله عنه)) said, "By Allah, we will prevent them." Ibn ‘Umar (رضي الله عنه) turned to him and told him off in an unprecedented fashion, saying: "I tell you what the Messenger of Allah (ﷺ) said, and you say, ‘By Allah, we will prevent them’!!" (Sahih Muslim 667)
‘Abdullah ibn ‘Umar (رضي الله عنه) reported: One of the wives of Umar ibn Al-Khattab (رضي الله عنه) attended dawn and evening prayers in congregation in the mosque. It was said to her, “Why do you go out while you know Umar dislikes it, and he is jealous?” She said, “What stops him from prohibiting me?” It was said, [what stops him is] “The statement of the Messenger of Allah (ﷺ): Do not prevent the maidservants of Allah from the mosques of Allah.” (Sahih al-Bukhari 858)
Conditions for permissibility
There are some conditions that need to be fulfilled, which are as follows:
1. Wearing proper hijab
2. Not going out wearing perfume
3. Having her husband's permission
And her going out to pray must not involve any other kind of haram activities, such as being alone in a car with a non-mahram driver.
Is a woman's prayer at home better than praying in the mosque?
However, it is also established in the authentic Sunnah that a woman's prayer at her home is better for her than going to the mosque to pray.
It was narrated that Umm Humayd (رضي الله عنها), the wife of Abu Humayd As-Sa`idi (رضي الله عنه), came to the Prophet (ﷺ) and said, “O Messenger of Allah, I like to pray with you.” He said, “I know that you like to pray with me, but your prayer in your room is better for you than your prayer in your courtyard and your prayer in your courtyard is better for you than your praying in your house, and your prayer in your house is better for you than your prayer in the mosque of your people, and your prayer in the mosque of your people is better for you than your prayer in my mosque.” So she issued orders that a prayer-place be prepared for her in the furthest and darkest part of her house, and she used to pray there until she met Allah (i.e., died).” (Musnad Imam Ahmad, 26550)
And Allah knows best.