Why did the Apostle Paul write 1st Timothy? What was the purpose and why is it still relevant today?
Paul’s argument to Timothy and the church at Ephesus is a clear explanation of the difference between the gospel of God being of grace and false ideas or teachings which are opposed to it. Paul tells how a person is justified by faith (1:15-16). This is opposed to the false teachings that the Law can redeem a person and put them into a right relationship with God (1:4-9, 4:1-5). The false teachers clash with the true doctrine that grace alone in faith alone in Christ alone saves (4:7, 10). Paul tells Timothy that it’s urgent to refute these falsehoods (1:3, 4) and explains that they are based upon human reasoning and not based upon the objective truth that is from God (1:6-7, 10). Paul gives further explanations as to how the church membership should behave (3:14, 15) in the event that he is delayed in coming (3:15). Paul argues from the standpoint that false teachers are bringing in human arguments and reasoning’s and that these false teachings are leading some astray (1:6, 4:1-3), just as they are doing today.
Instructions for the Church
The false doctrines at Ephesus are first introduced at the beginning of this book and there is a great sense of urgency (1:3-11). Then Paul’s doctrine of justification is juxtaposed to the false teachings to show how a person is truly justified (1:12-17). Paul gives Timothy direct instructions for how the church membership should act (2:1-3:16) and gives the proper format and reasons for prayer, both corporately and privately (2:1-8). He also addresses the proper role of women in the church, and by extension, in the home (2:9-15). He also lays out the qualifications for church leadership, probably due to the false teachers who are not qualified according to God (3:1-13). The proper conduct of widows and the elders, both at home and in the household of God, are given (5:1-21). Paul states his specific purpose for writing his letter to Timothy and the church, telling Timothy that this is “how one ought to behave in the household of God” (3:14, 15).
Paul next addresses how to deal with false teachers and how to identify them by what they teach, (4:1-3) and by what they forbid (4:3). This is in direct contrast to what God has said in His Word (4:4-16). Pastoral responsibilities come next with how Timothy should deal with members sinning openly in the church (5:1-2), how slaves should live as an example for the gospel (6:1-2), and what Timothy has to do behind the pulpit (5:6, 6:2, 13). He also urges Timothy to fight for the truth that Paul taught him (6:12- 13). A final admonition near the end of the letter is given to Timothy in how to deal with false teachings (6:3-5), the kinds of evil that the love of money brings (6:6-10), the character of a godly man of faith (6:11-16), and how to guard the objective truth of God given to Timothy’s charge (6:20, 21).
The book or letter is organized in such a way as to address the damage that false teachers have already done in the church at Ephesus (3:14-16, 4:1, 2). Paul begins the letter with warnings about false teacher’s teachings (1:3-11), covering the false teachers again in the middle of the letter (4:1-5), and ends the letter with yet another warning that concludes the letter (6:3-5). This is indicative of the reason that Paul wrote this letter. Paul is addressing the seriousness of the false teachings that have infiltrated the church through false teachers. The organization of the letter speaks in no uncertain terms that he is to take care of himself since these issues create great stress on the young pastor (5:22).
By the sheer volume and repeated uses of warnings about false teachers and their false teachings, which dominate the entire letter, these two terms alone are frequently emphasized: to watch for false teachers and their false teachings. They are repeated more than anything else in this letter. It was not only in Ephesus, but in nearly every church that Paul had established or wrote to. The Jews were following Paul around and bringing in elements of the Law which are not related to the gospel that Paul had taught Timothy and which is found in the New Testament churches (1:4, 9, 4:1-5). In fact chapter one, four, and six are primarily dedicated to the issue of false teachers who have brought in false teachings. Even the church leadership qualifications in chapter three were likely intended to ensure godly leaders would be installed in the church and that they were actually qualified so that they could oppose those false teachers who may have already been in positions of leadership (3:1-13). This makes sense because Paul gives Timothy the leadership qualifications (3:1-13), telling him that he hopes to “come to [Timothy] soon…so that if I delay, you may know how one ought to behave in the household of God (the church)…” (3:14-15).
I believe that Paul does move to a climax about the primary focus of this letter to Timothy. After two previous warnings about false teachers and their false teachings, and the result of these ungodly teachings (1:3-11, 4:1-16), he writes about those who have already been affected (6:4, 5, 9). This is particularly noted where Paul says that they “plunge people into ruin and destruction,” and that the love of power and money is at the “root” of the problem (6:9, 10). Paul concludes the letter by warning Timothy that their teachings are both godless and “swerve from the faith,” which is a dangerous position to be in if you’re a teacher (6:20).
Stirring to Action
There is a sense of urgency to keep false teachers from teaching their false teachings which are not what Paul personally taught Timothy (1:3). Paul also tells Timothy to expose them and restrain these false ministers from teaching their mythological, empty teachings (1:4). Paul insists that what they teach they don’t even understand, even though they are sure of themselves and their teachings (1:5). They are supposed to be teaching those things that are in “accordance with the gospel” (1:11), so Paul commands Timothy, like a general would a command someone, that he’s got a fight on his hands and so he needs to gird his loins with the truth (1:18-19). This type of warfare necessitates the dire need for prayer (2:1). To ensure that godly teachers are ordained in the church, Paul lists qualifications that cannot be compromised (3:1-13). Paul already knows that some will depart from the true gospel of God (4:1) by using doctrines that are straight out of hell and not from God; teachings that are not ordained by God (4:2-5). Paul gives this imperative command to Timothy to teach that our hope is only found in the Savior (4:10, 11), and to teach the congregation this same truth (4:13). 
Timothy should keep the church in line by making sure that those who are trying to take advantage of their position do not succeed. One example is to identify widows who are truly widows and those who are not, and for families to take care of their own household (5:3, 4, 8). Paul gives final instructions for Timothy, which is also the main theme of this letter to teach the truth from God (6:2), and stop any false teachings dead in its tracks (6:3). Finally, Paul admonishes Timothy to fight with all his might, to guard the truth like you would guard a treasure, which it is…the great and precious promises of God (6:2).
Here is some related reading for you: Timothy in the Bible: Character Profile 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible: English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.