table{border:1px solid #000;} doesn't seem to work without the border=1 statement.
In the same way as changing <input width=30> into input{width:400px;}, I would like to use <table> and declare the border in css only. Is that possible?
Update my mistake was using
table{border-width:1px;}
instead of e.g.
table{border:1px solid #000;} 
--which works fine.
Use this CSS:
table {
    border-collapse: collapse
}
td {
    border: 1px solid #000
}
Live Demo
border-collapse: collapse is important; without it you get doubled borders, like this.
Absolutely - that is the preferred way. You might have to style td as well as tr.
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