The topic for today is going to be one of those which different people will see in other ways. So let me begin by saying if you see it differently then, feel free to start a conversation in the comments.
I believe a person has no right to call themselves a Christian or a believer unless they are striving to live obediently by everything God has asked. This includes being a tither, seeing everything you own as His, and being generous with those in need.
Tithing is giving ten percent of your income, from every income source, to the local church God has led you to join. I think the tithe should be for the support of the local church. For this reason I do not believe giving to a designated account, or to a special offering, or to a para-church organization is tithing. But all of these forms of giving can be legitimate ways to give offerings above and beyond your tithe and this is a part of being generous.
Stewardship is a Biblical principle for believers. Everything we are, and everything we have belongs to God. We are to care for it and manage it like it is something owned, not by us, but by the Lord. The pretty part of this picture is we can trust Him to take care of us; we will never be abandoned. But the difficult part of this picture is that we need to curb our wants according to what He would want us to have and often deny ourselves in order to be money wise.
A part of stewardship is to care about the things that God cares about. Most notably this includes those around us who are in genuine need. This giving needs to be done under the direction of the Holy Spirit, because of the deceitful nature of our world. But when God says give, we should give. (When I say the world is deceitful I mean it is full of people who appear needy, but would actually use the money to harm themselves and others.)
Now let me also say, giving sometimes has its pitfalls. Pride gets into our giving just like it does into everything else. So some act as if their giving means they get to be arrogant or bossy around the church. Some people use their giving as a means to control the shape of ministry. This is not giving, this is manipulation. It is not giving to God as an act of trust, it is using money to exert personal power and put yourself in the place of God.
So give, but give with the right attitude.