Question: Should the church be led by Elders or by the Congregation?
Answer: The short answer to that question is simply "yes".** Scripture clearly teaches that the congregation must (a) submit to its leaders on many matters while also (b) requiring the congregation to make decisions on certain key matters.
There are of course people who deny either part (a) or part (b) of the above sentence. Those who deny part (a) hold to an absolute form of what is called Congregationalism. These individuals believe that the Congregation is to be the final court of appeal on all matters of the church. However, this blatantly ignores many New Testament teachings, including the clear command in Hebrews 13:17 for the congregation to submit to its elders. Furthermore, Scripture refers to elders as "overseers" and as those who "rule".
Conversely, there are those who deny part (b). This essentially is an absolute form of Elder-Rule which maintains that the congregation is not to vote on any issue, but rather the Elders alone are to make decisions (albeit with input from the congregation...maybe). What I find troubling with this approach is a similar blind-eye (or even hermeneutical twisting) of Scriptures clearly supporting congregational decision-making. The Bible tells us that elders are not to lead with coercion, but by persuading the congregation to freely follow (1 Peter 5:3). Second, it tells us that elders may be censured and are not above criticism of the Body (1 Timothy 5:19). Thirdly, two passages in Scripture depict the gathered church assembly as the decisive court of appeal in matters of church discipline. 1 Corinthians 5:4 directs charges to be brought before a sinning brother in front of the "assembled church" and in Matthew 18:15-20 Jesus himself directs the assembled church to pronounce judgement upon an unrepentant brother ("if he refuses to listen even to the church..."). We might also look to Acts 6 where the assembled church chose its leadership. While that may or may not be referring to the office of deacon, it clearly reveals that the NT church--at least on some occasions--had a decisional role on certain matters. There are numerous other examples. In Acts 1:15-26 Peter asks the assembled believers in Jerusalem (about 120 gathered brothers in the Lord) to pick a replacement for Judas. Notice he did not just have the Apostles make this decision, but rather all the believers who were assembled together. In Acts 15:22 it says that "it seemed good to the apostles and the elders, with the whole church, to choose men from among them and send them to Antioch with Paul and Barnabas." Also, in 1 Corinthians 14:29-30 Paul notes that the whole congregation is to weigh and discern the teachings of someone who purports to speak a prophetic revelation.
Absolute elder rule and absolute congregational government are not found in the New Testament. What we do find is the clear command to "submit to one another" (Eph 5:21). I fear that those advocating an 'absolute' form of church government do so (at the expense of clear biblical teaching) out of a refusal to submit. Sadly, we have congregations who refuse to submit to their biblical leaders and we have biblical leaders refusing to submit to their congregations. All of it is sin.
Scripture doesn't give us details. It does not provide of list of things the elders can decide on and a list of things on which the congregation must decide. It does give us (1) clear examples that the congregation made real decisions, (2) clear examples of the congregation being involved with the decisions of the elders/apostles, and (3) clear commands to submit to spiritual leaders. Some have termed this Elder-led Congregationalism.
** I am assuming that we all believe that Christ--and Christ alone--is the head of the church. I interpreted the question as asking about the human element and structure of church leadership.