The Greek noun ῥῆμα "saying, utterance, verb, word" is analyzed as consisting of the root ἐρ-/ῥε- (er-/rhe-) "say" (confer εἴρω "I say"; ἐρῶ "I will say") and the suffix -μα (-ma), a suffix used to form nouns from verbs. (However, according to a falsified but popular folk etymology, ῥῆμα is derived from the verb ῥέω "I flow".)
In the New Testament, this noun is used in such instances as the following: “τὸ δὲ ῥῆμα Κυρίου μένει εἰς τὸν αἰῶνα” “the Lord's utterance/saying remains forever” (1 Peter 1:25). Rhema "saying" in this context bears a different meaning than Logos "word".
From Strong's Exhaustive Concordance, Greek word #4487. It can be found in the following New Testament passages (not exhaustive):
Some Pentecostal Christians view Rhema as the voice of the Holy Spirit as he speaks to the believer at the present moment. In this sense, the Christian should be guided by the Holy Spirit as he/she is guided through inner feelings, impressions, dreams, visions, and signs. God's Rhema, the direct words of God to the individual, can also be imparted through the words of others, such as a preacher in a worship service, or a friend who counsels them. In this sense, God's direct guidance can be determined by a variety of means. The test of the authenticity of a Rhema from God is simple... How does it compare to the whole of Scripture? Orthodoxy says that God will not speak a word that contradicts His written word, the Scriptures, so there is a built-in safeguard to prevent misinterpretation.
By contrast, Logos is typically used in Scripture to refer to what God has said to His people, that is, the collection of God's sayings about Himself, His relationship with His creation and His Church. The Logos of God is true for all time and in all places. It is easy to see how this can be interpreted both as the bible (God's Word Written) and in the Person of Jesus Christ (God's Word in the Flesh) as seen in John's Gospel, Chapter 1.
The Holy Spirit often uses passages in the Logos of God to create specific guidance, Rhema, for the individual. An example of the Rhema of God is to consider how several different people can read the same passage of Scripture and each sees something different. This occurs because each person is in a different place spiritually in their individual lives so God's word to each of them in that moment of their lives is different - specific to their needs. The Logos (God's original meaning and intention) has not changed, but the Holy Spirit has used it to speak to each according to his or her needs
Some Non-Pentecostal Evangelical Christians, however, have a much different understanding of rhema. Many evangelicals see rhema as being almost synonymous with logos. In other words, the specific guidance we receive from the Holy Spirit at any given time can only be discerned by the general principles laid down in the written word - the Bible. But this sort of guidance is not the same as that described above - it can only give general guidance about how the decision should be made. If we used the same example of the Christian wondering about when to start their business, the evangelical would argue that God will not give any specific guidance - it is up to the person to work it out for themselves. What God can give guidance on is on how that business should be run, and what the business actually does (Running a Brothel, for instance, would be wrong).
For evangelicals, then, the rhema is when God speaks directly to a person as they apply the Bible's teaching to a specific subject. Some Evangelicals do not believe that the Christian may make up their own mind when there is no Biblical teaching on a situation. However, this will never come up because these Evangelicals believe that there is a biblical teaching on every situation, specific or otherwise.